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

AI Thread Summary
The discussion centers on proving the zero component lemma for antisymmetric 4-tensors, which states that if any off-diagonal component is zero in all inertial coordinate systems, then the entire tensor must be zero. Participants express confusion about extending the proof from 4-vectors to 4-tensors, particularly in how to mathematically express the relationship between off-diagonal components and the overall tensor. They reference the use of Lorentz transformations to demonstrate that if a component of a related 4-vector is zero, then all off-diagonal terms must also be zero. The conversation highlights the need for clarity in mathematical expression to solidify the proof. Overall, the thread seeks guidance on formalizing the proof for antisymmetric 4-tensors.
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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.


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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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