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 International System of Units is given by:

Maxwell's equations differential form

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 :

Lorentz force

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:

action variation
q is the electric charge.

The variation of the action gives:

variation of action

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

variation of action and integral

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:

potential variation

Which gives for the variation of the action:

variation action & potential

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

electromagnetic tensor

 

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

The electromagnetic tensor is antisymmetric:

antisymmetric tensor

And the diagonal components in it are equal to zero:

diagonal components

 

The components of this tensor can be found using:

potential and field definition

For example:

tensor components

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

electromagnetic tensor components

The components of the contravariant tensor can be found using:

contravariant tensor

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

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

tensor transformation law

The following equation :

Maxwell equations first group

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:

Levi-Civita tensor

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

Lagrangian

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:

Lagrangian equations
Thus Lagrange’s equations give the result:

Lagrange's equations Maxwell  result

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

Maxwell equations second group

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

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

Maxwell equations tensors comma notation

 

In a gravitational field the electromagnetic tensor is given by:

electromagnetic tensor gravitation

\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 :

first and second group Maxwell equations

 

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:

second group with gravitation

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  :

equations of motion

Or equivalently:

motion of partilce

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:

energy momentum tensor
And the Einstein field equations

Einstein field equations.

are called the Einstein-Maxwell equations.

To be continued…

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