Derivatives of EM Four-Potential: Euler-Lagrange to $\nabla \times B$

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In summary, the Euler-Lagrange equations give a relation between the components of the field tensor and the current density, but it cannot be directly converted to a relation between the field tensor and the current density. To do so, one must use the Minkowski metric and the permutation symbol to relate the field tensor to the electric and magnetic fields. Additionally, in the (1+3) formalism, the components of the field tensor are related to the electric and magnetic fields in a different way, requiring a conversion between temporal-spatial components and purely spatial components.
  • #1
Gene Naden
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So the Euler-Lagrange equations give ##\partial _\mu ( \partial ^\mu A^\nu - \partial ^\nu A^\mu ) = J^\nu## with ##B=\nabla \times A##. I want to convert this to ##\nabla \times B - \frac{\partial E}{\partial t} = \vec{j}##. I reckon I am supposed to use the Minkowski metric to raise or lower indices, but am not sure how. I want to get from ##(\partial ^\mu A^\nu - \partial ^\nu A^\mu ) ## to ##\nabla \times A##.
 
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  • #2
Gene Naden said:
I want to get from ##(\partial ^\mu A^\nu - \partial ^\nu A^\mu )## to ##\nabla \times A##.

You can't, at least not if ##\nabla \times A## is supposed to contain all of the information; ##(\partial ^\mu A^\nu - \partial ^\nu A^\mu )## is a 4-vector expression and ##\nabla \times A## is a 3-vector expression, so the first contains information that the second does not.

Try writing out the components of ##(\partial ^\mu A^\nu - \partial ^\nu A^\mu )## explicitly, making sure to include ##\partial_0## and ##A^0##. You should find that some of the components work out to ##\nabla \times A##, where ##A## is a 3-vector; but there are also other components that mean something different. Once you figure out what those other components mean, you will have the solution to your problem.
 
  • #3
PeterDonis said:
You can't, at least not if ∇×A∇×A\nabla \times A is supposed to contain all of the information; (∂μAν−∂νAμ)(∂μAν−∂νAμ)(\partial ^\mu A^\nu - \partial ^\nu A^\mu ) is a 4-vector expression and ∇×A∇×A\nabla \times A is a 3-vector expression, so the first contains information that the second does not.
Not only is it an expression using 4-vectors instead of 3-vectors. It is an expression that describes the components of a (2,0) tensor while ##\nabla \times \vec A## is a 3-vector. Yoy can use the permutation symbol to relate the field tensor to the 3-vectors ##\vec E## and ##\vec B##.
 
  • #4
Of course one can convert from the Minkowski-covariant tensor notation to the non-covariant (1+3) notation (in a fixed (!) inertial reference frame). Indeed, the contravariant spatial components of the Faraday tensor is directly mapped one to one to the (1+3) Cartesian components ##\vec{B}##. You have (latin indices run over the spatial indices only, i.e., ##j \in \{1,2,3\}## etc.):
$$F^{jk}=\partial^{j} A^{k} -\partial^{k} A^j=-\partial_j A^k + \partial_k A^j=\epsilon_{ikj} (\vec{\nabla} \times \vec{A})_i.$$
The usual difficulty is to keep in mind that in the (1+3) formalism
$$\vec{\nabla} = \vec{e}_j \frac{\partial}{\partial x^j}=\vec{e}_j \partial_j=-\vec{e}_j \partial^j.$$
One should also note that the notation is not easily made consistent since in the (1+3) formalism one usually writes all indices as lower indices, because in Cartesian components you have V_j=V^j, but in SR of course V_j=-V^j.

The difficulty is natural since the components ##\vec{E}## and ##\vec{B}## are vectors in the (1+3) formalism (i.e., their components behave as vector components under rotations in the fixed inertial frame), but they are not spatial components of four-vectors but in the 4-formalism are components of the antisymmetric Faraday tensor.

For completeness, here's the relation between the temporal-spatial Faraday tensor components with the (1+3) object ##\vec{E}## (electric field):
$$F^{j0}=\partial^j A^0-\partial^0 A^j=-\partial_j A^0-\partial_0 A^j=E_j,$$
i.e.,
$$\vec{E}=-\frac{1}{c} \dot{\vec{A}}-\vec{\nabla} A^0,$$
as is well-known from the (1+3) formalism of E-dynamics.
 

1. What are derivatives of EM four-potential?

The four-potential in electromagnetic theory is a mathematical construct that combines the electric and magnetic fields into a single entity. Derivatives of the four-potential refer to the rate of change of this entity with respect to time or space.

2. What is the significance of Euler-Lagrange in this context?

Euler-Lagrange equations are used to determine the functional dependence of a system on its parameters. In the context of EM four-potential, these equations are used to calculate the equations of motion for the system based on its Lagrangian function.

3. How are these derivatives related to Maxwell's equations?

Maxwell's equations describe the fundamental laws of electromagnetism. The derivatives of EM four-potential can be used to derive these equations, providing a mathematical framework for understanding the behavior of electric and magnetic fields.

4. What is the role of $\nabla \times B$ in this concept?

The curl of the magnetic field, $\nabla \times B$, is a fundamental quantity in electromagnetism. It is closely related to the derivatives of EM four-potential and plays a crucial role in understanding the behavior of magnetic fields in the presence of changing electric fields.

5. How do these derivatives relate to the propagation of electromagnetic waves?

Electromagnetic waves are a fundamental aspect of electromagnetism, and their propagation can be described using the derivatives of EM four-potential. These derivatives help determine the speed, direction, and behavior of electromagnetic waves as they travel through space.

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