Proving invariance of scalar product

[Gabor]

Hi everyone,

How would I go about proving that the scalar product of two four-vectors (A,B) is invariant under a Lorentz transformation?

 

[robphy]

As a warmup, you might try to prove that the scalar product of two vectors in the plane is invariant under a rotation.

 

[Gabor]

Okay... I could do that for 2 vectors (x1, x2) and (y1, y2) in a plane.

As for the four-vector proof, I'm not even sure I'm doing it right... My understanding is that I have to take the scalar product of the two vectors A and B. Then I have to apply Lorentz transform to both vectors and calculate the scalar product of A' and B'. For invariance, these two scalar products should be equal?

 

[robphy]

yes.

How did you do the problem for the dot product of vectors in the plane?

 

[Gabor]

I figured out the proof for the four-vectors. Now I see the similarity between that and the rotation proof. Turns out I was using the wrong transformation formulas for my vectors and that's why things didn't add up. Thanks for your help!