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We propose the generalization of Einstein’s special theory of relativity (STR). In our model, we use the (1 + 4)-dimensional space G, which is the extension of the (1 + 3)-dimensional Minkowski space M. As a fifth additional coordinate, the interval S is used. This value is constant under the usual Lorentz transformations in M, but it changes when the transformations in the extended space G are used. We call this model the Extended space model (ESM). From a physical point of view, our expansion means that processes in which the rest mass of the particles changes are acceptable now. In the ESM, gravity and electromagnetism are combined in one field. In the ESM, a photon can have a nonzero mass and this mass can be either positive or negative. It is also possible to establish in the frame of ESM connection between mass of a particle and its size.

We consider the Extended space model (ESM), which is a generalization of Einstein’s special theory of relativity (STR). The ESM is formulated in a 5-dimensional space, or more specifically in a (1 + 4)-dimensional space with the metric

The foundations and different properties of this theory are presented in Ref. [

In the STR, the rest mass m of a particle is a Lorentz scalar. For photons,

Such particle having a mass m, corresponds to a hyperboloid in Minkowski space, in the limiting case this hyperboloid degenerates into a cone.

Since the change of the mass of a particle corresponds its transition from one hyperboloid to the other, i.e. change of the corresponding interval, it seems natural to choose interval s as an additional fifth coordinate. Thus, we will work in a space with coordinates

We denote this space as

The usual (1 + 2)-dimensional cones and hyperboloids occur as sections of the surface (3) by hyperplanes

The 5-dimensional ESM has a number of advantages compared to STR. Firstly, it is a more symmetric theory. In this model, the energy, momentum and mass are equivalent and can be transformed into each other. Secondly, in this model there is no fundamental difference between massive and massless particles and they can be transformed into each other either. In addition, under the ESM electromagnetic and gravitational fields are combined into a single field. This field is investigated in papers [

In this work, we will show that it is possible to compare a nonzero mass to a system of photons, and that it is possible to establish connection between mass of a photon and its size.

The various aspects of the concept of “mass” in STR was discussed by Okun’ [

In Minkowski space

is associated to each particle (Ref. [

For free particles, the components of the vector (5) satisfy the equation

It is well-known relation of relativistic mechanics, which relates the energy, momentum and mass of a particle. Its geometric meaning is that the vector (5) is isotropic, i.e. its length in the space

We compere parameter

Here

A set of variables (5) forms a 5-pulse, its components are conserved, if the space

The gravitational effects in ESM were discussed in Ref. [

In the usual relativistic mechanics and field theory the mass of a particle is constant, and for particles with zero masses and nonzero rest masses different methods of description are used. The particles with nonzero rest masses are characterized by their mass m and speed

The 4-vector

corresponds to a particles with nonzero rest mass.

The 4-vector

corresponds to a particles with zero mass.

The length of the 4-vector

It follows from (12) that length of a massive vector (10)

In the frame of our approach, there is no difference between massive and massless particles, and therefore one can establish a connection between two methods of description of these two sorts of particles. This can be done using the relation (9) and the hypothesis of de Broglie, according to which these relations hold for the massive particles. Now, substituting (9) in (5), we obtain the relation between the mass m, frequency

It follows that if

Now we construct 5-vectors from 4-vectors (10), (11). We suppose that a 5-vector

corresponds to a stationary particle of mass m.

The 5-vector of a particle, which moves with velocity

Similarly the 4-vector (11) transforms into 5-vector

At the transition to a moving coordinate system the vector (17) does not change its form, only the frequency

Thus, in empty space in a stationary reference frame there are two fundamentally different object with zero and nonzero masses, which in the space of

and

The vector (19) describes a photon with zero mass, the energy

cribes a stationary particle of mass m. The photon has a momentum

tum equal to zero. In the 5-dimensional space, these two vectors are isotropic, in Minkowski space only the vector (19) is isotropic.

The length of the 5-vector

It follows from the definition (21) that 5-vectors (16), (17) are isotropic vectors, ie their length is equal to zero.

If we restrict ourselves to Lorentz transformations in Minkowski space it is impossible to transform an isotropic vector into anisotropic one and vice versa. In other words in frame of the SRT photon can not acquires mass, and a massive particle can not be a photon. But in the Extended space

As it was already mentioned the parameter n connects the speed of light in vacuum with that in the medium:

For hyperbolic rotation through the angle

As a result of this transformation a particle with mass is appeared.

The velocity of this particle is defined by formula (7).

Under the same rotation the massive 5-vector (20) is transformed as

Under such rotation a massive particle changes its mass

and energy but conserves its momentum.

The rotation through the angle

Given this, the photon acquires the mass

and velocity

The vector of a massive particle is transformed in accordance to the law

In this transformation the energy of a particle is conserved but its mass and momentum change

It is easy to see that vectors (23), (25) and (27), (30) are isotropic.

It is important that photon mass, which is generated by transformations (23), (27), can have either positive and negative sign. This immediately follows from the symmetry properties of

It is well known that one can compare a plain wave and vector (19) to a photon only in empty space. If there is some particle or field in the space in addition to initial photon it is necessary to describe this photon by other vector. In order to find this vector let’s consider the system of two photons with a same energy. We suppose that these photons are moving in the same plane but in different directions. In the empty space these photons are described by 5-vectors (19)

and

Here

The energy E of the system of two photons is

The momentum P of the system of two photons is

These photons do not interact with each other, therefor the system of two such photons is a free system and it must be described by an isotropic 5-vector. This vector reads

We see that one can associate with a system of two photons a mass

Here

Let us compare now formulas (27) and (36). We see that in our case the angle

The other approach to the problem of constructing a mass of a system of photons was proposed by Rivlin [

There is a natural way in the frame of ESM to establish a connection between mass of a particle and its size. It can be done with the help of an analogy between the dispersion relation for a free particle

and dispersion relation for a wave in the hollow metal waveguide

Here

The similarity of the ratios (37) and (38) drew the attention of many scientists. One can associated with the critical frequency

This parameter has the unit of mass, and the question arises, if this quantity can be interpreted as a real mass? The mass, which acquires the electromagnetic field when it enters the waveguide. In the works of Rivlin, this problem was studied in a systematic way [

That is the value, which we propose to consider the characteristic linear parameter that is associated with the particle.

The value (41) resembles the Compton wavelength of the electron, however, the physical meaning of it is very different. In the formula for Compton wavelength of the electron, the parameter m is the rest mass of the electron, but in Equation (41) m is the mass that a photon acquires when it is subjected by external influences.

In the empty space a free photon is described by a plane wave and has an infinite size. The mass of this photon is equal to zero, but its energy is finite. It is an idealized object. In does not exist in reality, because in reality there is no absolutely empty space. But when photon enters the space with external fields, it acquires a non-zero mass m. In accordance with the formula (40) a finite linear parameter l can be compared to this mass m. We consider the linear parameter l as a size of a photon. Such reduction of an infinite format of a free photon to finite size of a photon in an external field us a result of action of this field.

In ESM an external action is described by rotations in Extended space G(1,4). We have discussed above the rotations from the group L(1,4), and set how changing the mass of the photon at these turns. Because a linear parameter l expressed by the formula (41) using the mass of the photon, with its help it is possible to find the dependence of this parameter from the values define these rotations.

So, in the case of rotations in the plane (TS) dependence of the photon mass from the angle of rotation

In the case of rotations in the plane (XS) dependence of the photon’s mass is determined by the formula (27). With its help, we obtain an expression for the parameter l through the angle

The rotation angles

the mass

In given work, the generalization of Einstein’s Special theory of relativity is proposed. It is the (4 + 1)- dimensional Extended space model. It is shown that in the frame of this model, it is possible to compare the mass and size of the photon. In forthcoming works, we will discuss the problem of localization of massive particles.

V. A. Andreev,D. Yu. Tsipenyuk, (2016) The Mass and Size of Photons in the 5-Dimensional Extended Space Model. Journal of Modern Physics,07,1308-1315. doi: 10.4236/jmp.2016.711116