Calculate Eigenvalues of Electromagnetic & Stress-Energy Tensors

In summary, the conversation discusses how to calculate the eigenvalues of ##F_{ab} = \partial_a A_b -\partial_b A_a## and ##T_{ab} = F_{ac} {F_b}^c- (1/4) \eta_{ab} F^2 ## without using the characteristic polynomial. The physical meaning of these eigenvalues is also questioned. It is noted that the eigenvalues of a real antisymmetric matrix are imaginary and the eigenvalues of a real symmetric matrix are real. However, it is pointed out that neither ##T_{\mu\nu}## or ##F_{\mu\nu}## have eigenvalues as they are (0,2) tensors. The use
  • #1
ergospherical
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How can we (as nicely as possible... i.e. not via characteristic polynomial) calculate the eigenvalues of ##F_{ab} = \partial_a A_b -\partial_b A_a## and ##T_{ab} = F_{ac} {F_b}^c- (1/4) \eta_{ab} F^2 ## and what is their physical meaning?
 
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  • #3
I mean, first of all you would have to define what you mean by "eigenvalues". The entire concept of an eigenvalue is that you have an operator from a vector space to itself. As such, neither ##T_{\mu\nu}## or ##F_{\mu\nu}## have eigenvalues because they are (0,2) tensors. You can, of course, raise and lower an index using the metric if you have one, but then you no longer have a symmetric or anti-symmetric matrix.
 
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Related to Calculate Eigenvalues of Electromagnetic & Stress-Energy Tensors

1. How do you calculate eigenvalues of electromagnetic and stress-energy tensors?

To calculate the eigenvalues of these tensors, you first need to represent them in matrix form. Then, you can use mathematical methods such as diagonalization or the characteristic polynomial to find the eigenvalues.

2. Why is it important to calculate eigenvalues of these tensors?

Eigenvalues provide important information about the behavior and properties of electromagnetic and stress-energy tensors. They can help us understand the distribution of energy and momentum in a system, and can also be used to solve equations and make predictions about the behavior of electromagnetic fields and stress-energy in different situations.

3. What is the significance of the eigenvalues of these tensors?

The eigenvalues of these tensors represent the possible values of energy and momentum in a given system. They can also reveal important information about the symmetry and stability of the system, as well as its behavior under different conditions.

4. Are there any real-world applications of calculating eigenvalues of electromagnetic and stress-energy tensors?

Yes, there are many real-world applications of calculating these eigenvalues. For example, in physics and engineering, eigenvalues are used to analyze and design electromagnetic devices and materials, as well as to model and predict the behavior of stress and strain in structures.

5. Can the eigenvalues of these tensors change over time?

Yes, the eigenvalues of these tensors can change over time depending on the dynamics of the system. For example, in an oscillating electromagnetic field, the eigenvalues will change as the field strength and direction change. Similarly, in a system experiencing stress and strain, the eigenvalues will change as the forces and deformations change.

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