The equations of electrodynamics , tensors , and gravitation-First Part

The featured image above was made with Mathematica . It represents  the propagation of an electromagnetic wave.

Elements of this post were written for an answer I gave at quora.com , the question was about Maxwell’s equations
in tensor form. I have modified the answer and added material and equations related to General Relativity and gravitation.

Maxwell’s equations are the fundamental equations of classical electromagnetism and electrodynamics. They  can be stated in integral form , in differential form (a set of partial differential equations) , and in tensor form.

The conventional differential  formulation of Maxwell’s equations in the is given by:

In the above equations , E is the electric field (vector field) , B is the magnetic field ( pseudovector field) ,  ρ is the charge density , and  J is the current density.
ε0 is the Vacuum permittivity ,
and μ0 is the Vacuum permeability .

The second and third equations above form the first group of Maxwell’s equations (the generalized Faraday’s law of induction and Gauss’s law for magnetism) , and the the first and last equations above form the second group  (Gauss’s law and Ampere’s circuit law extended by Maxwell) .

The equations of electromagnetism (Maxwell’s equations and the Lorentz force) in covariant form (invariant under Lorentz transformations) can be deduced from the Principle of least action .

The electromagnetic wave equation can be derived from Maxwell’s equations , its solutions are electromagnetic (sinusoidal) waves.

All the steps and equations related to the topic of this post will not be shown here because they would  make the post much too long , but an overview of the equations , calculations and derivations  will be presented.

The development of the components of the Lorentz force and (after some calculations) its formulation in tensor form allows the introduction of the electromagnetic field tensor Fij :

where uj is the four-velocity four-vector .

The variation of the action δS with respect to the coordinates of a particle gives the equations of motion.

The action for a charged particle in an electromagnetic field can be expressed as:

q is the electric charge.

The variation of the action gives:

After some calculations and an integration by parts the variation of the action becomes:

In the equation above , ui is the four-velocity four-vector.

Since the trajectories of the particle are supposed to have the same initial and final coordinates , the first term in the right-hand side of the equality above is equal to zero.
The potentials are a function of the coordinates , and  the following equalities can be used:

Which gives for the variation of the action:

Thus the electromagnetic field tensor emerges  from the variation above ; in fact the electromagnetic tensor is given by the definition:

Ai is the electromagnetic four-potential comprising the electric scalar potential V and the magnetic vector potential A .

The electromagnetic tensor is antisymmetric:

And the diagonal components in it are equal to zero:

The components of this tensor can be found using:

For example:

The others components can also be calculated and one gets for the covariant electromagnetic tensor:

The components of the contravariant tensor can be found using:

where $g^{\text{ij}}$ is the Metric tensor.

The transformation laws for the electromagnetic tensor are given by the general relation:

The following equation :

gives the first group of Maxwell’s equations .

If the Levi-Civita tensor ( $e^{\text{ikmn}}$ ) is used ( a completely antisymmetric 4th rank tensor) , one gets for the first group of equations:

Using the expression of the action with the Lagrangian density and taking the variation of the action as stationary:

The last relation above represents the Lagrange equations. And the electromagnetic tensor  Fik is assumed to have been found and defined after developing the Lorentz force.

The relations above give:

Thus Lagrange’s equations give the result:

And the second group of Maxwell’s equations is given by :

The second group of equations also gives the equation of charge conservation :

Another notation (the comma notation) and an additional way to write Maxwell’s equations in tensor form are the following :

In a gravitational field the electromagnetic tensor is given by:

$\nabla _j$ is the covariant derivative and $\Gamma _{\text{ij}}^k$  are Christoffel’s symbols .
So the relation between the electromagnetic tensor and the potentials is unchanged in the presence of a gravitational field.

The first and second groups of Maxwell’s equations become :

After calculation the first group can be shown to be equal to the original first group of equations without gravitation.

The second group can be expressed as:

where g is the determinant of the metric tensor.

The equations of motion of a particle of charge q in the presence of an electromagnetic field and a gravitational field are given by  :

Or equivalently:

In General Relativity , the energy-momentum tensor Tij of an electromagnetic field in free space is expressed in the form of the electromagnetic stress- energy tensor:

And the Einstein field equations

are called the Einstein-Maxwell equations.

To be continued…