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The velocity of the electromagnetic radiation in a perfect dielectric, containing no charges and no conduction currents, is explored and determined on making use of the Lorentz transformations. Besides the idealised blackbody radiation, whose vacuum propagation velocity is the universal constant c, being this value independent of the observer, there is another behaviour of electromagnetic radiation, we call inertial radiation, which is characterized by an electromagnetic inertial density , and therefore, it happens to be described by a time-like Poynting four-vector field which propagates with velocity . is found to be a relativistic invariant expressible in terms of the relativistic invariants of the electromagnetic field. It is shown that there is a rest frame, where the Poynting vector is equal to zero. Both phase and group velocities of the electromagnetic radiation are evaluated. The wave and eikonal equations for the dynamics of the radiation field are formulated.

In his famous lecture delivered for almost 93 years now to the Nordic Assembly of Naturalists at Gothenburg [

So, the problem is to derive a formula for velocity of the electromagnetic radiation as a certain function of the field strengths [

The aim of the present paper is to elaborate a method for finding the velocity of the e-m radiation, corresponding to the transport of the energy, momentum and angular momentum, as a function of the field strengths.

Z. Oziewicz [

In this paper, we explore the dynamics of energy and momentum of the electromagnetic field by introducing the concepts of velocity of the radiation and the field mass density. Our method is based on the same idea of Z. Oziewicz proposed to identify the velocity of the radiation with the velocity of the system of reference, where the Poynting vector vanished. However, oppositely to our result, he concluded that there was no physical rest frame, with

In order to define a velocity for the electromagnetic radiation, a simple logic scheme has to be used: there exists a certain inertial reference system where the density of the momentum of the inertial e-m radiation field is equal to zero. The velocity of this inertial system is identified with the velocity of the radiation. So, according to this concept, in order to define the velocity of the field, it is sufficient to know the transformation laws of the field under the Lorentz-group.

The Concept of the Inertial Electromagnetic FieldIn the solution of any electromagnetic problem the fundamental relations that must be satisfied are the four field equations―Maxwell equations [

According to Maxwell theory the velocity of the electromagnetic (e-m) waves is defined via permittivity

The physical sense of the speed of the electromagnetic radiation is attached to the velocity of the flux of radiation transporting energy, momentum and angular momentum. Since the speed of the light in the medium is defined by formula

where

is the vacuum wavelength. We are addressing both cases occuring

and

It is seen, that in the medium the value of the e-m field characteristics change in such a way that the property of transversality keeps conserved, but the relationship

which is valid for the radiation field in the vacuum, in the medium holds no more true.

This argument is taken as a pivoting idea to introducing the concept of inertial e-m radiation field, as a transversal e-m field, which is principally characterized by the main condition

Once the concept of inercial electromagnetic field is defined, the rest of the paper is organized as follows. First we present an alternative covariant double-scalar potential formalism for the representation of transversal electromagnetic fields (Section 2). Thereafter we derive the formula for the phase velocity of the electromagnetic radiation field (Section 3). As an application of this formula, we write the eikonal equation of the geometrical optics, and the wave equation for electromagnetic radiation field for a definite energy density, Poynting vector and mass field density (Section 4). Finally, in Section 5 the main conclusions of the work are thrown.

In Ref. [

The Maxwell’s equations in the vacuum are equations for the electric field strength

As we very well know, the equation

is automatically satisfied, if the magnetic-flux density is represented as

because of the identity

The expression for the electric field strength is in turn expressed by

These formulae are such that the second group of Maxwell equations representing Faraday’s law and the absence of magnetic charges are automatically satisfied. The first group of Maxwell equations are reduced to the following two equations for the potentials

Equations (2.6a, 2.6b) can be separated by choosing the so called Lorentz gauge condition

Substitution of Equation (2.7) into (2.6a, 2.6b) yields wave equations for the four-potentials

Now, let us rewrite these formulae in their tensorial form. From the potentials

Further, let us introduce four-coordinates

In this notation, the vectors of electric field strength and magnetic flux density may be cast into the form of a screw-symmetric tensor

Then, formulae (2.4) and (2.5) are joined into one expression

The first group of Maxwell equations takes the form

whereas the second part of Maxwell equations admits the following form

In order to obtain the wave equation for the four-potential vector, usually, the Lorentz-gauge condition is used

With this condition, the Maxwell equations are reduced to the wave equation for the four-potential vector

With respect to Lorentz transformations the electromagnetic fields are characterized to possess two invariants

First of all let us notice that the equation

is satisfied by defining the magnetic-flux density as

For the electric field strength one obtains

In a tensorial form these formulas are given by

Obviously, the functions

or in tensorial notation

The advantage of using this two-potential representation resides in the fact that we automatically introduce the two desired degrees of freedom. The first main consequence of this approach is the property of transversality of the electromagnetic field. In fact, in this representation, the field strength vectors satisfy the equation

Notice, however, that the dealing of the second invariant

It is to be noticed, that by reducing the degrees of freedom from four to two, additional algebraic relations between the fields and the four potential vector arise, they are namely

What kind of interpretation can be given to these equations? Looking for an answer, we refer the reader to Ref. [

Furthermore, it is shown that for the static magnetic field

Or in the components

The four-vector

From this last equation it follows that

The Lorentz gauge Equation (2.12) appears written in our two-potential representation as

which can be evaluated to

This equation separates into two Klein-Gordon type equations

where the parameter

We may also define in our formalism the current density as

Furthermore, this current density satisfies the continuity equation

which arose above as the Lorentz-gauge condition (2.12) for the potential

Consider two observers

where

Suppose that the velocity of

Then formulae (3.1) are reduced to Heaviside formulae

These transformation formulae keep unchanged both invariants:

The Poynting’s vector is transformed as follows

Now suppose that in the

Then,

From this equation it follows that the velocity vector

In order to construct an equation for the unknown constant

Suppose that

Let us introduce the following quantities

and

Evidently,

Consequently,

which has two different roots

By using this quantities in (3.8) we obtain two kinds of velocities

and

The first velocity

These velocities are explicitly expressed via field strengths as follows

Thus, if

In this notation, the formula for the velocity is written as

From this formula it follows the expression for the density of the energy

and for the Poynting’s vector

Let

Then,

Comparing these formulae with the analogous formulae for the relativistic point-particle, it is seen, that in the case of e-m field, for theenergy and the momentum, the rapidity appears multiplied by the factor “2”.

We have derived two kinds of velocities

Consequently, we have

for

for

From these formulae it follows that the velocities

By analogy we may also define the group velocity V

satisfying always

The group velocity is related to the phase velocity by the formula

where v can be either

In the previous section we have obtained formulae for phase velocities as functions of the densities of the energy, momentum and the mass. Consider classical electromagnetic waves travelling according to a scalar wave equation of the simple form

where

By assuming

with

The solutions of the wave equation are looked of the form [

with real R and

In the limit of geometrical optics, according to the eikonal equation

The Eikonal equation in a geometrical wave theory has the form

For the rays inside the medium with refractive index n defined via

and

The concept of velocity for the radiation fields is a long stated problem. This problem of the velocity of the electromagnetic waves has been always considered as a definitely solved problem: these waves have a velocity equal to

which is used also to be written as

this expression has been considered phenomenological in nature, it is however relativistic. Here, it is clear that we are refering to the phase velocity of the waves, which can be both smaller as well as larger than the speed of light. However, we have been able to relate this velocity to the true physical group velocity of the radiation which is always present. We have developed this study in a relativistic formalism, considering the velocity of the electromagnetic radiation field as proposed by Zbigniew Oziewicz. This formalism is constructed out of first principles, because it is based on the transformations of Lorentz group, in order to derive the formula for the velocity of the reference system where the Poynting vector is equal to zero. In this way we arrived to the formula connecting the phase velocity with the densities of the energy and momentum.

From the viewpoint of the theory, we have been able to describe the inertiality of the electromagnetic Field, we are able to assure that with the exception of the propagation in vacuum, where the Poynting vector is a null four-vector, in a dielectric medium, the Poynting vector migrates into a time-like four-vector, whose norm is the inertial field density, formula (3.12), that is why the rest frame of the propagation is reachable. To the present, it has been known to us only the massless quantum mechanical behavior of photons. We don’t dare, for the moment, to describe them, inside matter, as mutants in a classical, massive state, but we just point out to the inertiality as an intrinsic hidden property of the electromagnetic field. How to quantify the inertial field density may surely be the subject of future works.

As we have introduced the notion of inertial field density as a measure of the inertia of the electromagnetic radiation, in return, we have formulated a new wave equation envolving the density of energy and the inertial field density.

Robert M. Yamaleev,A. R. Rodríguez-Domínguez, (2016) As Regards the Speed in a Medium of the Electromagnetic Radiation Field. Journal of Modern Physics,07,1320-1330. doi: 10.4236/jmp.2016.711118