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

I’ll start this continuation of the topic of a previous post by  an exposition and explanation of the concept of tensors in relativity physics  given by Albert Einstein , from his 1916 paper The Foundation of the General Theory of Relativity :

” The laws of physics must be of  such a nature  that  they apply  to systems of reference in any kind of motion.

The general laws of nature are to be expressed by equations which hold good  for all systems of coordinates, that is, are co-variant  with respect to any substitutions whatever (generally covariant).

Let certain things (” tensors “)  be defined with respect to any  system of  coordinates  by  a  number of functions of  the  coordinates , called  the  ” components ”  of the  tensor. There  are  then   certain  rules  by  which   these components can  be calculated   for  a new system of  coordinates , if they are known  for the original  system  of coordinates, and if  the  transformation  connecting  the two systems is known. The things  hereafter  called    tensors are further characterized by the fact  that the equations of transformation for their  components are linear  and  homogeneous.  Accordingly,  all  the components in  the  new system  vanish,  if  they all  vanish in the  original system. If,  therefore, a law  of nature is expressed  by equating all the components of a tensor to zero, it  is  generally covariant. By  examining the  laws of  the formation of  tensors, we acquire  the  means  of  formulating generally covariant laws. “

Source : The Principle of Relativity (Dover Books on Physics)
Einstein’s paper can also be found here:
The Foundation of the Generalised Theory of Relativity

I will deal in this post with the derivation of the Einstein field equations of General Relativity and with the related equations of motion.

As in the previous post , I will not show all the long and detailed calculations but I will give an overview of the equations , calculations and derivations .

We start with the derivation of the field equations free from the sources generating the field.The equations are deduced from the principle of least action.

A small remark : in the equations below , the symbol g written in two ways is the same letter in two different fonts , it represents either the metric tensor (with subscript or superscript) or the determinant of its matrix representation.

The action of the gravitational field is given by:

where χ is the gravitational constant appearing in Einstein’s field equations.
Ω is the product of the differentials of the coordinates.
In four-dimensional space we have :

 

In the action integral above , The scalar curvature (or the Ricci scalar) R is related to the Ricci tensor by the equation R = g^ij R_ij  .
And the Ricci tensor is related to the Riemann curvature tensor by the equation:

Hence we have:

The variation of the action yields:

After calculation the first integral in the variation of the action above gives:

The third integral in the variation of the action is equal to zero:

And the variation of the action can be written as:

The variations δg^ij are arbitrary and the Einstein field equations independent of the sources generating the field
(  R_ij – (1/2)g_ij R = 0 ) can be deduced from the integral above.

The action for matter and the electromagnetic field is given by:

This action and its variation involve the Lagrangian density:

After integrating and calculating we get:

Defining the energy momentum tensor (or stress–energy–momentum tensor ) by the relation:

We get for the variation of the action:

The preceding results lead to the condition δ(S_g+S_me)=0 being written as:

The equality above must remain valid for arbitrary variations of δg^ij  ,and we deduce :

which are the Einstein field equations of General Relativity.

The contravariant form of the energy-momentum tensor is given by the transformation :

Below is a schematic description of the contravariant components of the energy-momentum tensor (image from Wikipedia) :

Note that of we take the more general form of the action :

The field equations for the most general case would become:

where λ is the cosmological constant .

The Einstein field equations are in agreement with the conservation of the energy-momentum tensor , which means its divergence is null:

The energy-momentum tensor (mixed tensor) can be written as the sum of two terms:

The first one is of electromagnetic origin:

The second one involves the presence of matter:

The condition of null divergence can be written as:

After some calculations we get for the first term :

And for the second term in the null divergence equation:

By using the last two equations above , equation (1) can be written as:

which is the equation of motion of a continuous distribution of charge inside a gravitational field and an electromagnetic field.

The references and sources in my Science books page and the problem solvers page can be viewed for books which I have read , worked out or consulted and which are related to the topics in this post.
I found the book by Jean-Claude Boudenot ( Electromagnetisme et gravitation relativistes ) helpful while I was studying these topics in the past.

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:

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 F^ij :

where u_j 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 , u_i 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:

 

A_i 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  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 ( ) 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  F^ik 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:

 is the covariant derivative and   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 T^ij 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…