Proving Orthogonality of Vector w/ Schnutz Special Relativity Tensors

In summary: 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!
  • #1
19
1
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
 
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  • #2
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
robphy said:
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
[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
robphy said:
[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
Does this operator do what you want it to do, independent of signature convention?
Use it and see.
 
  • #7
robphy said:
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.
 
  • #8
fengqiu said:
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.
 
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  • #9
Orodruin said:
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
Do you mean Schutz, A First Course in General Relativity?
 

1. What is the Schnutz Special Relativity Tensor?

The Schnutz Special Relativity Tensor is a mathematical tool used in special relativity to describe the relationship between two reference frames. It is a 4x4 matrix that represents the transformation between space and time coordinates in the two frames.

2. How do you prove orthogonality of vectors using the Schnutz Special Relativity Tensor?

To prove orthogonality of vectors using the Schnutz Special Relativity Tensor, you must first calculate the inner product of the vectors in both reference frames. Then, using the tensor, you can transform the coordinates of one vector into the other frame and recalculate the inner product. If the result is zero, the vectors are orthogonal.

3. What is the significance of proving orthogonality of vectors in special relativity?

In special relativity, the concept of orthogonality is closely related to the principle of causality. If two vectors are not orthogonal, it means that they are not mutually exclusive and could potentially affect each other's measurements. By proving orthogonality, we can ensure that causality is preserved and that the laws of physics hold true in different reference frames.

4. What are some applications of proving orthogonality in special relativity?

Proving orthogonality of vectors in special relativity is essential in many areas of physics, including electromagnetism and quantum mechanics. It is also used in the development of theories such as the Lorentz transformation and the theory of relativity.

5. Are there alternative methods for proving orthogonality in special relativity?

Yes, there are alternative methods for proving orthogonality in special relativity, such as using the Lorentz transformation or applying the principle of causality directly. However, the Schnutz Special Relativity Tensor provides a more efficient and systematic approach to proving orthogonality in complex systems.

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