Fundamental Way of Charge Formation and Relation between Gravitational Field and Electromagnetic Field ()
1. Introduction
In the theory of relativity, mass velocity relation gives a large quantity of kinetic energy attributed to the particle of rigid configuration [1]. Again this expression (concept) would be same for a spinning particle of rigid configuration as in [2]. A particle can possess two or more superimposed motions to the view of an observer. In [3] Chandrau Iyer and G. M. Probhu describe a constructive method for the composition of two planar boosts with velocities
and
resulting in a velocity
. Composition of three linear velocities
,
and
in different directions resulting in a single velocity has been developed in [4]. This leads to the assumption that a body can possess three simultaneous superimposed rotational or spin motions. In this paper, first, trials would be made to derive the energy momentum characteristic of a particle having two superimposed motions. An electromagnetic mass was possible due to Einstein [5]. Einstein strongly believed that field is the form of energy momentum tensor. A charged particle contains electromagnetic field. Again, photons contain electromagnetic field and also electric charge as in [6,7]. In this paper trial would be made to find out the complex characteristic of photon and following this consideration trial would be made to find out the source of fundamental charge of an elementary particle. Gravity is the four dimensional interaction. It is the space-time curvature, basically stress energy tensor. The principles of General Relativity imply that gravity and electrodynamics affect each other. According to Einstein all forces of nature are rooted in gravity. He dreamed of unification in between gravity and electrodynamics in much the same way as electricity and magnetism had been combined by Maxwell and Lorenz. But his unified field theory was not particularly successful. Following the general relativistic approach unified field theory has been developed. In [8] it is said that change of both electric and gravitational field results in the creation of a magnetic field in the region of space time which has a dual electro-gravitational nature. Change in magnetic field results in the creation of both electric and gravitational field. Rotating gravitational masses generate magnetic field B and which generates electric field E and gravitational field G. In this work trial would be made to derive the unification of gravity and electrodynamics following the combination of electricity and magnetism.
2. Complex Momentum of a Particle
For composition of three velocities
we may visualize six inertial frames as discussed in [4] where, co-ordinate of an event from frame S to frame S5 would be
(1)
Inverse of this operation would be
(2)
Considering four inertial frames
where, velocity
and angle
in [4] then matrix
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
[3]. Then magnitude of resultant velocity of an event of
as observed by
would be
(5)
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 [4] then it may be called rotation-rotation interaction (i.e. R-R interaction) where, respectively
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 [9] this may be written as
(6)
So
(7)
where,
also,
and
are normal to each other.
represents velocity along space coordinate and
represents velocity along time coordinate. So, ![](https://www.scirp.org/html/5-4500067\3d4fbafb-013c-4f21-932f-896003b43a56.jpg)
in (7) is a four velocity and from it we get a four velocity matrix
where,
(7a)
Hence to an observer in
resultant velocity
in (7) is of complex nature. And the relativistic mass of the particle
(8)
and relativistic Lagrangian would be
(9)
which leads to the form of relativistic momentum
(10)
where,
,
and ![](https://www.scirp.org/html/5-4500067\edbbbd48-eb57-42ed-ae82-3fc1ccdfeff8.jpg)
when θ = 90˚, then we get from (7)
(11)
where,
are mutually normal to each other which leads to the form of momentum
(12)
where,
and
![](https://www.scirp.org/html/5-4500067\a80c226f-a594-461a-b4e2-3f7ed5d3bd0a.jpg)
It is seen that Equations (10) and (12) represents a complex momentum of the particle. For a relativistic particle taking
we get from (11)
(13)
For R-R interaction
,
and
.
For R-L interaction
and
(i.e. direction of linear velocity of
)
![](https://www.scirp.org/html/5-4500067\99ec80b4-93d4-44fe-b077-b82cb6af1b67.jpg)
So, following (12) momentum would be
(14)
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 [2]. So S-S interaction or S-L interaction or L-L interaction of the particle implies that
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
(15)
So, we can write a function of field
as below
(16)
where,
is canonically conjugate to ![](https://www.scirp.org/html/5-4500067\fecf8645-472c-428e-8ca1-a9f7d5d2cc6f.jpg)
here,
and
are real fields,
and
are constants.
3. Complex Momentum and Field of Photon
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
(17)
This leads to the electromagnetic wave function as specified in [12,13] respectively
(18)
(19)
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
(20)
where
is the Maxwell stress tensor [14]. Since
is the basis of electromagnetic field so,
in (18) would be four dimensional wave function as
(21)
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.
4. Field Transformation
Let gravitational field function in the frame
in Section 2 be
and which would be
as observed by ![](https://www.scirp.org/html/5-4500067\786a263a-4ff5-4d96-a9dc-857fa22a6b8b.jpg)
where ![](https://www.scirp.org/html/5-4500067\7c3e9fa3-816a-4b27-a7a4-34df7d0a9585.jpg)
Then using (3) and (4) we get the relation between
and
as
(22)
Inverse of this operation is given as
(23)
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
(24)
This leads to the form
(25)
Similarly, using four velocity matrix as in (7A) we can consider the above relation as
(26)
where
and
are two constants which depend on the medium.
5. Conclusion
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.
6. Acknowledgements
Author thanks the authorities of Satmile High School, Satmile-721452, W. B., India for their continuous encouragements.
NOTES