Is an Antisymmetric 4-Tensor Zero if Any Off-Diagonal Component is Zero?

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SUMMARY

The discussion centers on proving the zero component lemma for antisymmetric 4-tensors, specifically that if any off-diagonal component is zero in all inertial coordinate systems, then the entire tensor must be zero. The participants reference the behavior of 4-vectors under Lorentz Transformation (LT) to establish that a zero component in one frame implies the entire tensor is zero across all frames. The challenge lies in expressing this proof mathematically for antisymmetric 4-tensors, which involve complex relationships among their components.

PREREQUISITES
  • Understanding of antisymmetric tensors and their properties
  • Familiarity with Lorentz Transformation in special relativity
  • Knowledge of 4-vectors and their representation
  • Basic mathematical skills for tensor analysis
NEXT STEPS
  • Study the properties of antisymmetric tensors in detail
  • Learn how to apply Lorentz Transformation to tensor components
  • Explore mathematical proofs involving 4-vectors and their implications
  • Investigate the implications of the zero component lemma in various physical contexts
USEFUL FOR

Students and researchers in theoretical physics, particularly those focusing on tensor calculus and special relativity, will benefit from this discussion.

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Homework Statement


prove the zero component lemma for any anti-symmetric 4-tensor: If anyone of its 0ff-diagonal component is zero in all inertial coordinate system, then the entire tensor is zero.


Homework Equations





The Attempt at a Solution



in case of 4-vector, if a particular component is zero in all inertial frame then by Lorentz Transformation in different direction, it can be proved that the 4-vector is zero in all inertial frame.
Here, i m confusing in how to prove it in case of anti-symmetric 4-tensor

Any help would be highly appreciated. thank
 
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I have the same problem. No answers yet.
off diagonal components of the antisymmetric 4 tensors in special relativity involves 3 vectors and we can form 4 vectors from them. If any component of that 3 vector is zero under LT the 4-vector is zero then all the off-diagonal terms are zero. This is what I thought but how can I express this in Mathematical Language?
If I'm wrong can you give me a clue about it?
 

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