A complex motion and complex momentum due to relativistic phenomenon has been deduced in this paper. This procedure leads to explain the generation of a field which is the result of energy momentum complexity (tensor). In this work, a form of complex momentum of photon has been derived. This momentum reveals the construction of electromagnetic field. These procedures have been applied to explain the electromagnetic field of fundamental charged particle and leads to the assumption of fundamental charge. In this works trial would be made to derive a relation between gravitational field and electromagnetic field.
In the theory of relativity, mass velocity relation gives a large quantity of kinetic energy attributed to the particle of rigid configuration [
and 
resulting in a velocity
. Composition of three linear velocities
, 
and 
in different directions resulting in a single velocity has been developed in [
For composition of three velocities 
we may visualize six inertial frames as discussed in [
Inverse of this operation would be
Considering four inertial frames 
where, velocity 
and angle 
in [
respectively turn out as
(3)
(4)
Above conditions imply that Frames 
and 
have both their co-ordinate axes aligned and 
is moving at a velocity 
along 
axis as observed by
. The inertial frame 
has another co-ordinate reference frame
, where 
axis of
, is rotated by an angle 
counter clockwise with respect to 
on 
plane. Frames 
and 
have both their co-ordinate axes aligned and 
is moving at a velocity 
along 
axis as observed by 
[
as observed by 
would be
In (5) it is clear that 
and 
are two linear velocities in different directions and rotational angle between 
and 
is
. From these implications we get Case-1: When 
and 
as in [
and 
are angular velocities of 
and 
about 
and 
axes. It is also pointed out that origin of frames are same with respect to
. If the particle is imagined at the origin of frame 
then it possesses two superimposed spins i.e. spin-spin interaction (S-S interaction) which is homogeneous with R-R interaction.
Case-2: When 
but 
is linear then it may be called rotation-linear (i.e. R-L) interaction and that will be S-L interaction if the particle is present at the origin of
.
Case-3: When 
and 
both are linear motions then it is called L-L interaction.
Following [
So
where, 
also, 
and 
are normal to each other.
represents velocity along space coordinate and
represents velocity along time coordinate. So, 
in (7) is a four velocity and from it we get a four velocity matrix 
where,
Hence to an observer in 
resultant velocity 
in (7) is of complex nature. And the relativistic mass of the particle
and relativistic Lagrangian would be
which leads to the form of relativistic momentum
where, 
, 
and 
when θ = 90˚, then we get from (7)
where, 
are mutually normal to each other which leads to the form of momentum
where, 
and
It is seen that Equations (10) and (12) represents a complex momentum of the particle. For a relativistic particle taking 
we get from (11)
For R-R interaction
, 
and 
.
For R-L interaction 
and 
(i.e. direction of linear velocity of
)
So, following (12) momentum would be
It is understood that due to every relativistic motion kinetic energy respective virtual mass 
is attributed to the particle and due to every relativistic spin it rotates about the axis with relativistic velocity approaching that of light [
performing two superimposed motions with velocity as in (13) reveals one kind of stress energy tensor (i.e. one kind of field). Following Equation (14) we get momentum-density
So, we can write a function of field 
as below
where, 
is canonically conjugate to 
here, 
and 
are real fields, 
and 
are constants.
Light is electromagnetic wave. It carries electric and magnetic fields which is proved by Faraday effect and Kerr effects. But photon is a particle whose kinetic energy is
, having mass
. Photon has spin motion about an axis and it may be considered as a small mass 
concentrated in a ring of radius 
and rotates at velocity of light 
and it also has linear motion with velocity 
along the axis of rotation [10, 11]. It is one kind of S-L interaction which is homogeneous with R-L interaction. So resultant velocity would be as in (13) which gives the complex momentum of photon as shown below
This leads to the electromagnetic wave function as specified in [12,13] respectively
where
, Now we can write 
and
where, 
and 
are respectively the momentum density energy density and poynting vector of electromagnetic field of photon, So we can write stress energy tensor of this field as
where 
is the Maxwell stress tensor [
is the basis of electromagnetic field so, 
in (18) would be four dimensional wave function as
Again from [6,7,15] a concept is that photon charge is possible with both types, positive and negative, and also upper limit of photon charge is 
of elementary charge. It is also possible that photon has two types of spins (i.e. clockwise or anti-clockwise). Since directions of field depend on the direction of momentum so, nature of charges (i.e. positive or negative) would be determined by the type of spins. It is understood that 
reveals the field. Photon carries electric and magnetic fields which are functions of energy 
(or virtual mass
) and momentum as given in Equation (17). From the concept of photon we can write energy-momentum tensor due to S-L interaction of a particle which appears as an electromagnetic field. From this we can assume that tensor due to S-S interaction of the particle generates electromagnetic field and the particle would be a rest charged element to the view of an observer in S. So a particle of rigid configuration may be charged having electromagnetic field if it possesses S-S or S-L interaction with corresponding relativistic speed. It is to be pointed out that electron, proton carry electric charge as well as electromagnetic field with its rigid configuration. Hence we can write, 
in (16) is an electromagnetic field function and to make a rest charged particle S-S interaction of it is required.
Let gravitational field function in the frame 
in Section 2 be 
and which would be 
as observed by 
where 
Then using (3) and (4) we get the relation between 
and 
as
Inverse of this operation is given as
It is understood that a particle of rigid configuration may be charged having S-S or S-L interaction with relativistic speed as in [2,10]. So we can write 
performing two superimposed motions (which is associated with
) generates electromagnetic field function 
which would be homogeneous with 
in (22). So we can consider a relation
This leads to the form
Similarly, using four velocity matrix as in (7A) we can consider the above relation as
where 
and 
are two constants which depend on the medium.
A particle can possess two simultaneous superimposed spins (i.e. S-S interaction). To get an electromagnetic field, energy-momentum tensor due to complex motion as well as complex momentum of a relativistic particle as in (7) and (10) are required. But to make an elementary rest charged particle, to the view of an observer, S-S interaction of it is required. In such manners a particle of rigid configuration having gravitational field generates electromagnetic field. Equation (26) reveals the relations between electromagnetic field and gravitational field.
Author thanks the authorities of Satmile High School, Satmile-721452, W. B., India for their continuous encouragements.