Proving invariance of scalar product

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SUMMARY

The discussion centers on proving the invariance of the scalar product of two four-vectors (A, B) under Lorentz transformations. Participants emphasize the necessity of applying the correct transformation formulas to both vectors and calculating the scalar product of the transformed vectors (A' and B'). The conversation also references a preliminary exercise involving the invariance of the scalar product of two vectors in a plane under rotation, highlighting the similarity in approach between the two proofs.

PREREQUISITES
  • Understanding of four-vectors in the context of special relativity
  • Familiarity with Lorentz transformations
  • Knowledge of scalar products and their properties
  • Basic concepts of vector rotations in a two-dimensional plane
NEXT STEPS
  • Study the properties of Lorentz transformations in detail
  • Learn about the mathematical formulation of four-vectors
  • Explore proofs of invariance for scalar products in various coordinate systems
  • Investigate the relationship between rotations in two dimensions and Lorentz transformations
USEFUL FOR

This discussion is beneficial for physics students, educators, and researchers focusing on special relativity, particularly those interested in the mathematical foundations of four-vectors and Lorentz invariance.

Gabor
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Hi everyone,

How would I go about proving that the scalar product of two four-vectors (A,B) is invariant under a Lorentz transformation?
 
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As a warmup, you might try to prove that the scalar product of two vectors in the plane is invariant under a rotation.
 
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?
 
yes.

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

Thread 'EPR revisited'

In an inertial frame of reference (IFR), there are two fixed points, A and B, which share an entangled state $$ \frac{1}{\sqrt{2}}(|0>_A|1>_B+|1>_A|0>_B) $$ At point A, a measurement is made. The state then collapses to $$ |a>_A|b>_B, \{a,b\}=\{0,1\} $$ We assume that A has the state ##|a>_A## and B has ##|b>_B## simultaneously, i.e., when their synchronized clocks both read time T However, in other inertial frames, due to the relativity of simultaneity, the moment when B has ##|b>_B##...
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