Proving invariance of scalar product

Click For Summary
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
Messages
3
Reaction score
0
Hi everyone,

How would I go about proving that the scalar product of two four-vectors (A,B) is invariant under a Lorentz transformation?
 
Physics news on Phys.org
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!
 

Similar threads

Graduate Show Lagrangian is invariant under a Lorentz transformation without using generators
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Lorentz transformation on mode functions
  • · Replies 7 ·
Replies
7
Views
1K
Graduate Principal Invariants of the Weyl Tensor
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Lorentz-Invariant Scalar Products
  • · Replies 17 ·
Replies
17
Views
4K
Undergrad Wave Cuadrivector: Proving Four-Vector Invariance of Light Velocity
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad A question about static spacetime in GR
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Meaning of the invariants built from the angular momentum tensor
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad How do Tensors "work" in relation to linear algebraic objects?
  • · Replies 7 ·
Replies
7
Views
902
Undergrad Einstein & Relativity: Metric, Tensor Fields, Scalar Product
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Why are scalar fields Lorentz invariant?
  • · Replies 11 ·
Replies
11
Views
5K