This paper research is the first part of the scientific theory that seeks to unify the sciences of physics with the minimal number of mathematical formulas as possible. We will prove that all equations of forces in nature can be concised in two mathematical formulas, no difference between gravitational or electrical forces or any other type of Types of conventional forces, and through the equivalence of the concepts of matrix and vector, in this theory we will be linking the four-dimensional forces equations with the classical physics as an introduction to connect the rest of the physical sciences.
Numerous of recent books in physics discourse the summarizing of Maxwell’s four equations into two equations, without addressing the possibility to generalize this concept to the rest of the forces, and work to link them with classical physics, which is the goal of publishing this research.
Whereas we cannot link all the physical sciences in one theory, unless the base upon which this theory was built represents a good and common ground for all of these sciences, So at the beginning will get to know some mathematical concepts (for example the relationship between the matrix and the vector) in a new and concise manner, with the remarks that we will deliberately ignore some proofs and details of those concepts to shorten the pages of this research that will exceed tens of pages.
In this part of the theory we will prove that all equations of force in nature belong basically to two basic mathematical formulas, on the
In the next parts we will explore the possibility of combining the theories of General Relativity and Quantum Mechanics with this theory in the minimal mathematical relationships as possible.
However, the purpose of publishing this paper can be summarized as follows
1) Introducing new mathematical ideas and concepts which will help to unify physics;
2) Unification of the physical sciences with as few equations as possible (in this part, most of the forces known as only two forms);
3) Linking modern physical science with ancient physics without resorting to any hypotheses (such as the stability of the speed of light in the theory of relativity).
In this paper, the space
where
In general, the space of two vectors
“The cross product of a set of vectors in any specified space equal to the Determinant of these vectors”
It mean that if
The main determinant of (m + 1) × (n) matrix has been defined. Denote by
where
m: the number of the vectors in the group;
and the dual sub determinant
where
Let,
Thus,
where,
► For example:
let,
Then, from above equation we get
The magnitude or length of the vector
From dot product, we get
The last equation equals the length of the vector in
From (2.1), we can define the dual vector
where,
From (2.1), we therefore get
► Now in the example at hand, we have
Let,
Then, the cross-product of two vectors
On the other hand, we have from vector relations the equation
Here the last vector equals the Equation (1).
The dot-product of
From (3.2), we therefore get
Here the new vector appearing on the right-hand side equals the Equation (II).
On A.E filed, there are only two types of forces namely cross and dot forces
Let
Therefore
Then
According to the three-orthogonal vectors e1, e2, e3 we can rewrite the field vectors as
Now consider the equations
Thus, the Equation (4.1) becomes
we therefore get
Let
From Equation (3.2) we have
where
Using the three-orthogonal vectors e1, e2, e3 we can rewrite the transformation in Equation (4.2) and Equation (4.3) as
then we obtain
Thus
For the orthogonal unit vectors e1, e2, e3 the last equation becomes
where
► Let in our example
Then
thus from Equation (4.4) and Equation (4.5) we get
The last two equations equals the Equation (I) and the Equation (II).
The 4-momentum P of a particle of mass m0 at position
velocity
The 3-velocity v of the particle is defined by
where
թ is the 4-momentum of the coordinate system itself, թ = (թ0, թ)
Thus
Now, we can rewrite last equation as the following
where
If the Equation (5.1) is equivalent to the cross force equations in (4.4), we shall have
From comparison above, we have
From Equation (5.3) then,
Return above we have in the three-dimensional space
So, in the four-dimensional space time we Consider the
For E M case, Let A is the vector potential and թ = qA [
The sub determinant of angular velocity
From Equation (5.3) and Equation (4.3) we then get
which is equivalent to Equation (5.2).
where ∅ is scalar potential energy, so we can write the 4-coordinate momentum թ as,
For 4-vector potential A, we get [
We suppose that the dual force
where
In A.E space, all force equations [
Using the transformation in Equation (4.2) and Equation (4.3), we obtain
The Equation (6.3.1) and Equation (6.3.2) follow that
By assuming that
and so on. The overall result is [
By a similar argument, we can write the dual matrix as
Without the component
to get the first E M Lorentz force law. Let
It follows that
Without the component
to get the second E M Lorentz force law. Let
It follows that
According to the equations above, we can define 4-Maxwell’s Equations by suppose that
By vector triple product we have
But
From Equation (3.3) thus
I have named the new space in this paper as A.E (Abou Layla-Erdogan’s) as an expression of my thanks and appreciation for the Turkish President’s humanitarian attitudes towards my people and appreciation for my Turkish friends that supported me during my high study in Turkey.
Abou Layla, A.K. (2018) Old Mechanics, Gravity, Electromagnetics and Relativity in One Theory: Part I. Journal of High Energy Physics, Gravitation and Cosmology, 4, 529-540. https://doi.org/10.4236/jhepgc.2018.43031