Property antisymmetric tensors

In summary, the conversation discusses how to prove a property of two antisymmetric tensors and whether a certain relation holds true. The attempt at a solution presents a possible method, but the incorrect balancing of indices is pointed out.
  • #1
phoenixofflames
5
0

Homework Statement



I was wondering how I could prove the following property of 2 antisymmetric tensors [tex]F_{1\mu \nu}[/tex] and [tex]F_{2\mu \nu}[/tex] or at least show that it is correct.

Homework Equations


[tex]\frac{1}{2}\epsilon^{\mu \nu \rho \sigma} F_{1\rho \sigma}F_{2\nu \lambda} + \frac{1}{2}\epsilon^{\mu \nu \rho \sigma} F_{2\rho \sigma}F_{1\nu \lambda} = - \frac{1}{4} \delta^{\mu}_{\lambda} \epsilon^{\rho \sigma \alpha \beta} F_{1 \alpha \beta}F_{2\rho \sigma}[/tex]

The Attempt at a Solution


If \mu = \lambda, the left side gives [tex]- \epsilon^{\rho \sigma \alpha \beta} F_{1 \alpha \beta}F_{2\rho \sigma}[/tex] and the right side also ( summing over \mu )

But how can I see that if \mu is different from \lambda, that this relation is true? The right hand side is zero, but how can I proof that the left side is also zero? I have no idea..
 
Last edited:
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  • #2
The indices are incorrectly balanced, there is a [tex]/nu[/tex] on the LHS but none on the RHS
 
  • #3
hunt_mat said:
The indices are incorrectly balanced, there is a [tex]/nu[/tex] on the LHS but none on the RHS

The \nu is contracted, only \mu and \lambda are free indices.
 
  • #4
Solved.
 

1. What is a property antisymmetric tensor?

A property antisymmetric tensor is a type of tensor in mathematics and physics that satisfies the property of being antisymmetric. This means that for any two indices of the tensor, swapping the indices results in a negative sign. In other words, the value of the tensor is equal to the negative of its own transpose.

2. What is the significance of property antisymmetric tensors?

Property antisymmetric tensors have important applications in various fields of mathematics and physics, such as differential geometry, electromagnetism, and general relativity. They are particularly useful in describing the behavior of vector fields and rotations in three-dimensional space.

3. How are property antisymmetric tensors different from other types of tensors?

Unlike symmetric or mixed tensors, which have the property of being unchanged when indices are swapped, property antisymmetric tensors have the opposite behavior. This makes them useful for describing phenomena that exhibit rotational or skew-symmetric behaviors.

4. Can property antisymmetric tensors be visualized?

Yes, property antisymmetric tensors can be visualized in the form of matrices or arrays, depending on the number of dimensions involved. For example, a 3x3 property antisymmetric tensor can be represented as a matrix with 9 elements, where the diagonal elements are zero and the off-diagonal elements follow the antisymmetric property.

5. Are there any real-world applications of property antisymmetric tensors?

Yes, property antisymmetric tensors have various applications in real-world problems such as fluid dynamics, electromagnetism, and mechanical engineering. For example, they are used to describe the rotation of rigid bodies and the motion of fluids in turbulent flow. They are also used in computer graphics to represent rotations and orientations of objects in 3D space.

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