1. Introduction
In my research “Old Mechanics, Gravity, Electromagnetics and Relativity in One Theory: Part I”, I published in “Journal of High Energy Physics, Gravitation and Cosmology (JHEPGC)”, I put some principles for a new theory in mathematics “titled The Extended Fields Theory”, where I derived some of the physics equations that can be applied to the electric and gravitational domains without distinguishing in this theory, which means that there is similarity between the Maxwell and Lorentz equations for the electromagnetic and gravitational fields, and to generalization this principle. I published two other research explained in the first that perihelion precession, deflection of light passing near to the sun, and the black holes, are cosmic phenomena which can be theoretically proved through classical method through symmetry principle between the electronic and gravitational fields, far away from the concept of space-time [1] . In the second, we obtained a precise ideal value of the universal gravitational constant [2] . The significance of this law lies in the fact that, it connects three different physical disciplines together, which are mechanics, electromagnetism and thermodynamics [3] without the concept of space-time.
Generally, in this new theory, we can prove that the cross product of two vectors in the
is directly proportional to the famous Einstein’s field equations in General relativity theory, where it is important for the unification of relativity with quantum mechanics, as well as also important for understanding of the universe, especially the metrics of general relativity (i.e. Reissner-Nordström metric), and the separation between the space-time structure and the big bang, because that prove there was a creator before the universe existed
As a special case, we will briefly in this paper to link the New Theory with Maxwell stress tensor equation, and in the next research, we will link Einstein’s tensor and quantum mechanics to our New Theory.
This research paper is part of several scientific research groups with different names to unite the science of physics to understand this scientific area; reader has to check the indicated references.
2. Electromagnetic Stress-Energy Tensor on A.E Filed
As shown in my last paper [4] , we can rewrite the cross force
for the electromagnetic field, and Conversion the mix cross-product of two vectors
and
to mix dot-product as
(2.1)
where
: 4-electric current density
: 4-E M Lorentz force density
Also we may rewrite the
for
as,
thus
where:
is the EM Maxwell stress vector, or electromagnetic stress vector, defined by
► For example:
let, qi is a complex orthogonal unit vector, and
As explained in my last paper we get
Thus
where
The magnitude or length of the Maxwell stress vector
is defined as
Then,
(2.2)
We can calculate the absolute value of the stress vector
by using the classical way as following,
thus
Then,
(2.3)
►► The last equation equals the length of the Maxwell stress vector in Equation (2.2).
3. The Value of Lorentz force Density by Matrix Form
Let us suppose the equation,
(3.1)
is the matrix form of stress vector, or electromagnetic stress tensor, defined by
We may conversion the mix cross-product of
and
to mix dot-product as shown in the master form bellow
(3.2)
where:
M: orthogonal vector,
for the electromagnetic field we have
Then we have
So,
Or as the following
(3.3)
► Now in the example at hand, suppose that
or as the form,
By rewriting the magnetic field vector
as the matrix in the form
we therefor get
(3.4)
On the other hand, we can conversion the square of vector
to matrix form as,
Then we obtain
Thus
(3.5)
►► Here the last force density equals the cross force as shown in Eq. (3.4).
4. Calculate the Stress Vector
In our theory we assume that
On the other hand, from master form and the results above, we have
Let, matrix A = (kμr)2: Mμr = −λI
Then we get
Here the Lambda symbol λ denote as Eigenvalue of matrix
,λ equals the average of the elements on the main diagonal of matrix A as follows:
where:
tr(A): is the trace of a matrix A (or the sum of the elements on the main diagonal of A)
I: unit matrix,
► For an example at hand, we will use the matter case where its assumed that,
Then we get
Taking the square of last equation
The eiginvalue λ can be easily defined by equation.
The cross-product of
and
is defined
Then we obtain
4.1. Calculate the Components of
by Master Form
► From last equation, we can now defined the components of the stress tensor as
(4.1)
(4.2)
(4.3)
4.2. Calculate the Components of
by General Relativity Way
► As shown in my last paper and example above, we have
Let,
By Comparing the middle-hand side with the right-hand side of the above equation we can be easily calculated the electric and magnetic field
In general relativity the components of the stress tensor shown as
Then we get,
(4.4)
(4.5)
(4.6)
►► The last Equations (4.4)-(4.6) are equivalent to the equations in (4.1)-(4.3).
This proves that there is no contradiction between this paper and the theory of general relativity, and this in no way happens if there is any mistake in new theory.
5. New Coordinate System
of the Stress Vector
5.1. By Using New Coordinate System
We Can Rewrite the Stress Vector
as the Following
►
where
is a orthogonal unit vector
5.2. The Length of the Stress Vector
► The length of the stress vector
is defined by,
Then,
(5.1)
►► Note that the last equation equals the Equation (2.2)
6. General Stress Tensor
6.1. Electromagnetic Stress Tensor
As shown in the example above, in the Extended Fields Theory, the cross-product of two electromagnetic vectors
and
can be written as the matrix in the form
Therefore
6.2. Generalization the Stress Tensor
In general, the cross force
can be written as
,
so
Suppose now that the vector
is defined by equation
Then
The last equation can be written as the
Or on the tensor form
(6.1)
6.3. Gravitational Stress Tensor
As it is known there is similarity between electromagnetic and gravitational laws, for example, the symmetry between the Maxwell equations for the electromagnetic (EM) fields and Maxwell equations for the gravito electromagnetic (GEM) fields, and the symmetry between the Lorentz’s force-law for EM fields and the Lorentz's force-law for GEM fields [5] .
so, we can rewrite the Equation (2.1) as the following
where
m: The masses of the objects.
: 4-mass current density.
: 4-GEM Lorentz force density.
: 4-Gravitomagnetic field.
: 4-Gravitational field.
The striking analogy between the Coulomb’s electrostatic force and the Newtonian gravitostatic force suggests that the analogous quantity for electrical permitivity
in gravitation is [6] :
we can deduce by analogy with electromagnetism that:
Now, we can rewrite the
for
as,
thus
where:
G: the gravitational constant.
is the gravitational permittivity of free space.
is the gravitational permeability of free space.
is the Maxwell gravitational stress vector, or Gravitoelectromagnetism gravitational stress vector, defined by
For now let us assume that,
Then
The last equation can be written as the
Or on the tensor form
(6.2)
The matrix
is the same as the Einstein’s tensor
in general relativity without space-time structure, this is what we will prove it in the next research
7. Conclusion
We do not need to prove the theory of general relativity or the theory of Big Bang to new physical hypotheses that are difficult to imagine, as the curvature of space-time induced by the presence of the massive body, as simple as it is, there is a creator for universe.
Acknowledgements
I thank the Amir of Qatar Sheikh Tamim bin Hamad Al-Thani for his humanitarian attitudes towards my people and his help in building my city.