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Journal of Computational Methods in Sciences and Engineering 01/2013; 13:59-109.
Alexander D. Dymnikov.
Louisiana Accelerator Center, Physics Department, Louisiana University at Lafayette, Lafayette, LA, USA.
Abstract
A matrix theory of n-dimensional mathematical field and the motion of mathematical points in n-dimensional metric space is developed. Two spaces are considered: the n-dimensional space of an integrable coordinate vector x with the integrable metric g(x) and the n-dimensional space of a non-integrable but differentiable coordinate vector {Q}, where dQ = edx, dτ ^2 = d\tilde {Q}dQ = d\tilde {x}g(x)dx, g(x) = \tilde {e}e, \partial (Q) = e^{ – 1}\partial (x). We call the coordinate space of the vector {Q} as the absolute space.
The derivatives of the non-integrable but differentiable matrix e are expressed through the elements of the Christoffel symbols and the elements of the Ricci and Riemann curvature matrices. The absolute velocity vector u(Q) \equiv {dQ}/{dτ} and the absolute mathematical field matrix P \equiv u(Q)\tilde {\partial}(Q) are introduced. We obtain two groups of the matrix field equations, the first of which is written in the two following forms: \partial (Q)\left\langle P \right\rangle – \tilde {P}\partial (Q) = \rho u(Q), Ku(Q) = \rho u(Q), where \left\langle P \right\rangle the trace of the matrix P, K is is the absolute Ricci matrix function, u (Q) is the n-dimensional absolute velocity vector, ρ is a scalar function, which is the eigenvalue of K with the corresponding eigenvector u (Q). The interpretation of this pure mathematical theory in 4-dimensional space is the theory of the electromagnetic and gravitational fields and the motion of charged and neutral particles in the electromagnetic-gravitational field.
Additional Information
Main results of the matrix field theory.
1. The novel, pure mathematical, matrix theory of the field in n-dimensional metric space is developed.
2. The physical results are the interpretation of the mathematical results.
3. The definition of the mathematical field matrix and the equations of motion of the mathematical point are given.
4. It is shown that the equations of motion are different for symmetric and antisymmetric field matrices.
5. It is proved that in the 4-dimensional metric space the field matrix is the electromagnetic-gravitational field matrix, where the antisymmetric part of the field matrix is the matrix of electromagnetic field and the symmetric part is the gravitational field matrix.
6. The elements of all obtained matrix functions are the Christoffel symbols of the first and the second order or their derivatives which are used in the tensor theory of relativity.
7. It is shown that the metric matrix is determined by the eigenvalues of the matrix, where g is the metric matrix and R is the Ricci matrix. There are two different types of the metric matrix equations and the marix equations of the field.. In the first type all eigenvalues of the matrixare equal and the appropriate eigenvector is arbitrary. We take it equal to the velocity vector. In the second type for only one eigenvalue the appropriate eigenvector is equal to the velocity vector.
8. The solution of the first type metric matrix equations in 4-dimensional spacetime gives the metric matrix with three different constants. Two constants are radius of one mass (charge) and the mass (charge) density. They both produce the curvature of the space. The second mass (or charge) increases the curvature, produced by the first mass (charge), and decreases the curvature, produced by the mass (charge) density.
9. The equations of motion in these metric space describe the motion of one mass (or charge) in the field of another mass (or charge) and mass (or charge) density.
10. The obtained second type of matrix field equations was solved also for the Friedmann-Lobachevsky model of the four-dimensional metric matrix g. We obtained the expression for the density of particles the comparison of which with the cosmic microwave background spectrum from COBE shows that it can be used as the mathematical model of the Big Bang.
Alexander D. Dymnikov, “Mathematical matrix theory of the field …with application in electromagnetic-gravitational fields.” http:// arxiv.org/abs/1211.4603
Figure legend
The comparison of our mathematical model
with the cosmic microwave background spectrum from COBE.
The Figure shows two curves. The continuous line is our function for d=5.4. The dotted line is the cosmic microwave background spectrum from COBE, where ρ is the intensity and s is wave lengths/centimetre [5].
