sign data derivative recovery

Journal of Computational Methods in Sciences and Engineering 13 (2013) 59–109 59DOI 10.3233/JCM-120454IOS Press

The matrix theory of mathematical field andthe motion of mathematical points inn-dimensional metric space

Alexander D. DymnikovLouisiana 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 metricspace is developed. Two spaces are considered: the n-dimensional space of an integrable coordinate vector {x} with the inte-grable metric g(x) and the n-dimensional space of a non-integrable but differentiable coordinate vector {Q}, where dQ = edx,dτ 2 = dQdQ = dxg(x)dx, g(x) = ee, ∂(Q) = e−1∂(x). We call the coordinate space of the vector {Q} as the absolutespace.

The derivatives of the non-integrable but differentiable matrix eare expressed through the elements of the Christoffel symbolsand the elements of the Ricci and Riemann curvature matrices. The absolute velocity vector u(Q) ≡ dQ/dτ and the absolutemathematical field matrix P ≡ u(Q)∂(Q) are introduced. We obtain two groups of the matrix field equations, the first ofwhich is written in the two following forms: ∂(Q) 〈P 〉 − P ∂(Q) = ρu(Q), Ku(Q) = ρu(Q), where 〈P 〉 the trace of thematrix 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 in4-dimensional space is the theory of the electromagnetic and gravitational fields and the motion of charged and neutral particlesin the electromagnetic-gravitational field.

Keywords: Alternative theory of gravitation, gravitational field, electromagnetic field, motion of planets

PACS: 04.20.-q; 04.20.Cv; 04.20.Jb

1. Introduction

The Nobel Prize winner E. Wigner writes in his article “The Unreasonable Effectiveness of Math-ematics in the Natural Sciences”: “The first point is that the enormous usefulness of mathematics inthe natural sciences is something bordering on the mysterious and there is no rational explanation forit. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of theuniqueness of our physical theories.”

Here we develop the matrix theory of n-dimensional mathematical field and the motion of mathemat-ical points in this field. This theory is a pure mathematical theory and for every application we need tointerpret the obtained results.

∗Corresponding author: Alexander D. Dymnikov, Louisiana Accelerator Center/Physics Dept/, Louisiana University atLafayette, 320 Cajundome Blvd., Lafayette, LA 70506, USA. Tel.: +1 337 482 1845; Fax: +1 337 482 6190; E-mail:[email protected].

1472-7978/13/$27.50 c© 2013 – IOS Press and the authors. All rights reserved

AUTH

OR

COPY

60 A.D. Dymnikov / The matrix theory of mathematical field and the motion of mathematical points

Two spaces are considered: the n-dimensional space of an integrable coordinate vector {x} with themetric g(x) and the n-dimensional space of an non-integrable but differentiable coordinate vector {Q}with the metric g(Q), where dQ = edx, g(x) = ee, and g(Q) = I .

Here e is a non-integrable but differentiable frame matrix. The derivatives of the matrix function eare expressed through the Christoffel matrices, which elements are the Christoffel symbols of the firstand the second kind, and through the Ricci and Riemann curvature matrices, which elements are theelements of Ricci and Riemann curvature tensors. We call the coordinate space of the non-integrablebut differentiable coordinate vector {Q}, as the absolute space. The absolute velocity vector u(Q) ≡dQ/dτ , where dτ ≡ √

dxg(x)dx =√

dQdQ, and the absolute field matrix P ≡ u(Q)∂(Q) areintroduced. We obtain two groups of the matrix field equations, the first of which is written in the twofollowing forms: ∂(Q)〈P 〉 − P ∂(Q) = ρu(Q), and Ku(Q) = ρu(Q), where 〈P 〉 is the trace of thematrix P ,

K ≡ ([DD]e)e−1 ,

[DD]ij ≡ DiDj −DjDi, i, j = 1, 2, . . . , n.

Dk ≡ ∂k(Q) =∂

∂Qk

Here K is the absolute Ricci matrix function, u(Q) is the n-dimensional absolute velocity vector, ρ isa scalar function, which is the eigenvalue of K with the corresponding eigenvector u(Q). Applicationsto the four-dimensional Riemannian space-time are considered. It is shown that in this case the absolutefield matrix can be considered as the absolute electromagnetic-gravitational field matrix. The symmetricpart of this matrix can be considered as the absolute gravitational field matrix. The antisymmetric partcan be considered as the absolute electromagnetic field matrix. The differential equations for these twomatrices and for the metric matrix were obtained. The partial case of these equations is the Maxwellequations for the electromagnetic field.

Four general solutions for the static spherically symmetric gravitational field are found. Two thesesolutions have no an apparent singularity at the Schwarzschild radius which the Schwarzschild metrichas and they correspond to the constant invariant density which is not zero (the Schwarzschild metrichas zero density). For the first solution, which contains the parameter, which can be interpreted as onemass, the formulae for the minimum and maximum orbital velocity and for the perihelion precession ofplanets through perihelion and aphelion distances are given. In our calculations of extreme velocities ofplanets all first four digits coincide with the NASA data. The second solution is more general solutionand it contains two parameters, which can be interpreted as two masses: one mass is moving in thefield of another mass. The obtained field equations were solved also for a maximally uniform space. Weobtained also the expression for the density of particles which looks like the expression describing themathematical model of the Big Bang, where the density at the beginning of time is zero and after that itis growing very fast until very big maximum and after this maximum it is slowly decreasing.

2. Notations and definitions. Vector and matrix functions and differential operators

2.1. Definition and notation of a matrix

A rectangular array of m×n numbers or scalar functions, set out in m rows and n columns, is a matrixof order m by n, or m × n. The numbers or functions aij are the elements of the matrix a, aij being

AUTH

OR

COPY

A.D. Dymnikov / The matrix theory of mathematical field and the motion of mathematical points 61

the element in the i-th row and j-th column. The row-subscript i ranges over the values 1, 2, . . . ,m, thecolumn-subscript j over the values 1, 2, . . . , n. The matrix as a whole is denoted by a, where

a ≡

⎛⎜⎜⎝

a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . .am1 am2 . . . amn

⎞⎟⎟⎠ .

We will write the matrix a also in a horizontal alignment (notations from Mathematica by S. Wolfram):

a ≡ {{a11, a12, . . . , a1n}, . . . , {am1, am2, . . . , amn}}.If m = n, a matrix a is a square matrix. The inverse matrix of a is denoted by a−1.

2.2. Transposing a matrix

From a m×n matrix a we may construct a new n×m matrix the rows of which are the columns of a,and consequently the columns of which are the rows of a. This operation is transposition, the resultingmatrix is the transpose of a and is denoted by a, where

a ≡ {{a11, a21, . . . , an1}, . . . , {a1m, a2m, . . . , anm}}. ≡

⎛⎜⎜⎝

a11 a21 . . . an1a12 a22 . . . an2. . . . . . . . . . . .a1n a2n . . . ann

⎞⎟⎟⎠

The matrix of order 1 × 1, that is, of one row and one column, is a scalar.

2.3. The trace of a matrix

The trace of a matrix a is denoted by 〈a〉,〈a〉 ≡ aλλ = a11 + a22 + . . .+ ann.

2.4. Two types of vectors: Coordinate vector as contravariant vector and differential operator vectoras covariant vector

2.4.1. Coordinate vectors x and QWe understand the material point as the material object with the point size, which is described by

the set of n parameters (or coordinates). The position (the set of n parameters) of any material point isdescribed by an n-coordinate vectors {x} or {Q} where

{x} = {x1, x2, . . . , xn}, {Q} = {Q1, Q2, . . . , Qn},dQ = edx,

e is a matrix function.Column vectors are written as

x ≡ {{x1}, {x2}, . . . , {xn}} ≡

⎛⎜⎜⎝

x1x2. . .xn

⎞⎟⎟⎠ , Q ≡ {{Q1}, {Q2}, . . . , {Qn}} ≡

⎛⎜⎜⎝

Q1

Q2

. . .Qn

⎞⎟⎟⎠ ,

and row vectors as

x ≡ {{x1, x2, . . . xn}} ≡ (x1, x2, . . . , xn), Q ≡ {{Q1, Q2, . . . Qn}} ≡ (Q1, Q2, . . . , Qn)

AUTH

OR

COPY


2.4.2. Differential operator vectors ∂(x) and ∂(Q)For the n-vectors {x} and {Q} the differential operator n-vectors {∂(x)} and {∂(Q)}{∂(x) ≡ (∂1(x), . . . , ∂n(x)), {∂(Q) ≡ (∂1(Q), . . . , ∂n(Q))

can be written as the column vectors ∂(x) and ∂(Q)

∂(x)≡{{∂1(x)}, . . . , {∂n(x)}}=⎛⎝ ∂1(x)

. . .∂n(x)

⎞⎠ , ∂(Q)≡{{∂1(Q)}, . . . , {∂n(Q)}}=

⎛⎝ ∂1(Q)

. . .∂n(Q)

⎞⎠,

or as the row vectors ∂(x) and ∂(Q),∂(x) ≡ {{∂1(x), . . . , ∂n(x)}} = (∂1(x), . . . , ∂n(x)),

∂(Q) ≡ {{∂1(Q), . . . , ∂n(Q)}} = (∂1(Q), . . . , ∂n(Q)),

where

∂i(x) ≡ ∂

∂xi, ∂i(Q) ≡ ∂

∂Qi.

2.5. The absolute coordinate vector, the absolute frame matrix and the absolute coordinate space [1]

The connection between the full (or exact) differentials dQ and dx is defined by the expressions:dQ = Q∂(x)dx = edx e = Q∂(x)

dx = x∂(Q)dQ = e−1dQ e−1 = x∂(Q)

dQdQ = dxeedx

ee = g(x)

where e is –a non-integrable but differentiable frame matrix, g(x) is the metric matrix of the coordinatespace x. We call the coordinate vector Q as the absolute coordinate vector and the frame matrix e as theabsolute frame matrix.

The non-integrable but differentiable absolute frame matrix e can be written in the form

e = Ωe0,

where e0 is an integrable frame matrix and Ω is a non-integrable but differentiable orthogonal matrix,

ΩΩ = In.

Any n× n matrix function F is integrable, if

(∂α∂β − ∂β∂α)F = 0, α, β = 1, . . . , n.

The integrability of the matrix e0 means that

(∂α∂β − ∂β∂α)e0 = 0.

From the expression for the infinitesimal invariant interval,

δτ2 = dxg(x)dx = dxeedx = dxe0e0dx,

it follows that

δτ2 = dQdQ = dQg(Q)dQ,

where the metric matrix g(Q) is the unit matrix In. We call the space of the non-integrable but differen-tiable absolute coordinate vector Q as the absolute coordinate space.

AUTH

OR

COPY


2.6. The contravariant and covariant vectors

The invariant scalar differential operator d can be written as the scalar product

d = ∂(Q)dQ = ∂(Q)edx = ∂(x)dx = ∂(x)e−1dQ.

From the last expression we will find the differential operator n-vector ∂(Q),

∂(Q) = e−1∂(x), ∂(x) = e∂(Q).

The vectors dQ and dx,

dQ = edx, dx = e−1dQ,

are the contravariant vectors and the vectors ∂(x) and ∂(Q) are the covariant vectors. The relation

dQ = edx

links the contravariant coordinate vector dQ and the contravariant coordinate vector dx. The relation

∂(Q) = e−1∂(x)

connects the covariant coordinate vector operator ∂(Q) and the covariant vector operator ∂(x).

2.7. Matrices formed by the absolute contravariant coordinate vector Q and the covariant vectoroperator ∂(x)

The expression Q∂(x), where Q is an n-vector function of x,

Q∂(x) ≡ {{∂1(x)Q1, . . . , ∂n(x)Q1}, . . . , {∂1(x)Qn, . . . , ∂n(x)Qn}},stands for the matrix whose ik-th element is equal to ∂Qi/∂xk . The transposed expression ∂(x)Q de-notes the matrix

∂(x)Q ≡ {{∂1(x)Q1, . . . , ∂1(x)Qn}, . . . , {∂n(x)Q1, . . . , ∂n(x)Qn}},whose ik-th element is equal to ∂Qk/∂xi. The absolute frame matrix e is the matrix

e ≡ Q←→∂ (x)

with the elements

eik = ∂Qi/∂xk.

2.8. The contravariant and covariant indices

In the matrix theory the indices are usually not shown. If it is necessary the following notations formatrix or vector indices are used: k – covariant index, k – contravariant index. Some vectors and matriceswith indices are shown below.

∂m(x) = ∂m(x),

xm = xm,

(Q∂(x))mn = (Q∂(x))mn =∂Qm

∂xn.

The matrix e = Q∂(x) is a contravariant-covariant matrix.

AUTH

OR

COPY


2.9. Short operator notations

In many cases we will use the short notations,

∂ ≡ {{∂1}, . . . , {∂n}}, ∂ ≡ {{∂1, . . . , ∂n}}, ∂k ≡ ∂k(x) =∂

∂xk.

D ≡ {{D1}, . . . , {Dn}}, D ≡ {{D1, . . . ,Dn}}, Dk ≡ ∂k(Q) =∂

∂Qk

Q∂ ≡ {{∂1Q1, . . . , ∂nQ1}, . . . , {∂1Qn, . . . , ∂nQn}}, Qi∂k ≡ ∂Qi/∂xk,

∂Q ≡ {{∂1Q1, . . . , ∂1Qn}, . . . , {∂nQ1, . . . , ∂nQn}}, ∂iQk ≡ ∂Qk/∂xi.

The scalar differential operator is d, where

d ≡ ∂dx = dx∂ =∂

∂xαdxα, d ≡ DdQ = dQD =

∂

∂QαdQα

The summation is understood to be carried out over any repeated Greek index symbol that appearstwice in any expression.

2.10. Unit vectors and matrices

We introduce the contravariant unit vector

i(λ) ≡ ∂λx,

covariant unit vector

i(λ) ≡ ∂xλ

and the unit vector i(λ), where

ij(λ) = δjλ.

If we are not interested in the nature of the unit vector, we use the unit vector i(λ).As an example, for n = 4

i(1) = {1, 0, 0, 0}, i(2) = {0, 1, 0, 0), i(3) = {0, 0, 1, 0}, i(4) = {0, 0, 0, 1}.From the contravariant and covariant unit vectors it is easy to obtain four different unit matrices,

i(λ)i(λ), i(λ)i(λ), i(λ)i(λ), i(λ)i(λ).If we are not interested in the nature of the unit matrix, we use the following definition of the unit (or

identity) matrix:

In = i(μ)i(μ) = i(1)i(1) + i(2)i(2) + . . .+ i(n)i(n).

As an example, for n = 4,

I4 = i(1)i(1) + i(2)i(2) + i(3)i(3) + i(4)i(4) =

⎛⎜⎜⎝1 0 0 00 1 0 00 0 1 00 0 0 1

⎞⎟⎟⎠ .

AUTH

OR

COPY


2.11. The antisymmetric differential compatibility operator matrix

The antisymmetric differential operator matrices [∂∂] and [DD],

[∂∂] ≡ {{[∂∂]11, [∂∂]12, . . . , [∂∂]1n}, . . . , {[∂∂]n1, [∂∂]n2, . . . , [∂∂]nn}},[DD] ≡ {{[DD]11, [DD]12, . . . , [DD]1n}, . . . , {[DD]n1, [DD]n2, . . . , [DD]nn}}

which are called the compatibility operator matrices, are defined by the elements

[∂∂]ij ≡ ∂i∂j − ∂j∂i, i, j = 1, 2, . . . , n.

[DD]ij ≡ DiDj −DjDi, i, j = 1, 2, . . . , n

The compatibility operator matrices may be written in the form

[∂∂] = [∂∂]αβi(α)i(β).

[DD] = [DD]αβi(α)i(β)

2.12. Two forms of the compatibility conditions

The vector function Q has the full (or exact) differential dQ,

dQ = edx, e = Q∂, Q∂i(j) = ∂jQ = ei(j)

which satisfies the compatibility conditions for the vector Q,

[∂∂]kjQ = (∂k∂j − ∂j∂k)Q = ∂kei(j) − ∂jei(k) = 0, k, j = 1, . . . , n.

The vector function x has the full (or exact) differential dx,

dx = e−1dQ, e−1 = xD, xDi(j) = e−1i(j) = Djx

which satisfies the compatibility (or integrability) conditions for the vector x

[DD]kjx = (DkDj −DjDk)x = Dke−1i(j) −Dje

−1i(k) = 0, k, j = 1, . . . , n.

Therefore there are two forms of the compatibility conditions:

[∂∂]kjQ = (∂k∂j − ∂j∂k)Q = ∂kei(j) − ∂jei(k) = 0, k, j = 1, . . . , n

and

[DD]kjx = (DkDj −DjDk)x = Dke−1i(j) −Dje

−1i(k) = 0, k, j = 1, . . . , n.

For the operator [DD]mμ, using the compatibility conditions

∂kei(j) − ∂jei(k) = 0, k, j = 1, . . . , n

and the connection between the covariant coordinate operator vector ∂ ≡ ∂(x) and the absolute operatorvector D,

D = e−1∂, Dm = i(m)e−1∂ = e−1mλ∂λ,

AUTH

OR

COPY


one obtains

[DD]mμ = DmDμ −DμDm = (Dme−1μλ )∂λ − (Dμe

−1mλ)∂λ + e−1

mαe−1μβ [∂∂]αβ = e−1

mαe−1μβ [∂∂]αβ ,

or, in the matrix form

[DD] = i(α)i(β)([DD]αβ = e−1[∂∂]e−1 = e−1i(α)i(β)([∂∂]αβe−1).

The last expression is the transformation from the twice covariant coordinate differential operatormatrix [∂∂] to the absolute twice covariant differential operator matrix [DD]. The expression

[∂∂] = e[DD]e

is the transformation from the absolute twice covariant differential operator matrix [DD] to the twicecovariant coordinate differential operator matrix [∂∂].

We consider a vector x as an integrable vector and a vector Q as an nonintegrable vector.

3. Derivatives of the non-integrable but differentiable frame matrix

The absolute non-integrable n× n frame matrix e satisfies the compatibility conditions [1],

∂jei(k) = ∂kei(j), j, k = 1, . . . , n ,

Dje−1i(k) = Dke

−1i(j), j, k = 1, . . . , n ,

where the matrices ∂je,Dje, j = 1, . . . , n are one-index matrices and where

ee = e0e0 = g(x).

3.1. One-index matrices

Let us consider an one-index n× n matrix function hc = {hcab}. This matrix can always be written asthe sum of two matrices with different symmetry,

hc ≡ γc + λc, hcab = γcab + λcab ,

with

λcab ≡

1

2(hcab − hbac + hacb − hbca + habc − hcba),

and

γcab ≡1

2(hcab + hbac + hbca − habc − hacb + hcba).

Here γcab has symmetry in c and b indices,

γcab = γbac ,

AUTH

OR

COPY


and matrix λc is an antisymmetric matrix,

λc = −λc, λcab = −λc

ba .

Consider the one-index matrix function hc, where

hc ≡ e∂ce, c = 1, 2, . . . , n, hcab =1

2(i(a)e∂cei(b)),

and e is the matrix function. Present the matrix function hc as the sum of two matrix functions withdifferent symmetry,

hc = γc + λc,

where the element γcab has symmetry in c and b indices,

γcab ≡1

2((e∂ce)ab + (e∂be)ac + (e∂be)ca − (e∂ae)bc − (e∂ae)cb + (e∂ce)ba),

and the matrix function λc is an antisymmetric matrix,

λcab ≡

1

2((e∂ce)ab − (e∂be)ac − (e∂be)ca + (e∂ae)bc + (e∂ae)cb − (e∂ce)ba)

hcab =1

2(i(a)e∂cei(b))

hcab − hbac =1

2(i(a)e∂cei(b) − (i(a)e∂bei(c)) =

1

2i(a)e(∂cei(b)− ∂bei(c)) = 0

Remembering that the symmetric metric matrix function g can be written as

g(x) = g = ee,

we find the matrices γc, λc and hc become

γc =1

2(∂cg − ∂i(c)g + gi(c)∂)

γcab = i(a)γci(b) =1

2(∂cgab − ∂agcb + ∂bgac)

λc =1

2(e∂ce− (∂ce)e+ ∂i(c)g − gi(c)∂)

λcab = i(a)λci(b) =

1

2(eaμ∂cgμb − (∂ceaμ)eμb + ∂agcb − ∂bgac)

hc =1

2(∂cg + e∂ce− (∂ce)e)

hcab = i(a)hci(b) =1

2(∂cgab + eaμ∂ceμb − (∂ceaμ)eμb)

The expression for the antisymmetric matrix λc can be written also in the form:

λcab =

1

2(i(a)(e(∂cei(b) − ∂bei(c)) − i(c)e(∂bei(a) − ∂aei(b)) + i(b)e(∂aei(c) − ∂cei(a)))

AUTH

OR

COPY


3.2. The one-index Christoffel matrix of the first kind

In the above equations the matrix elements γcab are the Christoffel symbols Γa,cb of the first kind,

γcab = i(a)γci(b) =1

2(∂cgab + ∂bgac − ∂agbc) = Γa,cb.

We call this matrix γc as the one-index Christoffel matrix of the first kind. Using the contravariant andcovariant indices we can write,

γcab = γcab.

The equationshc = e∂ce = γc + λc

γc =1

2(∂cg − ∂i(c)g + gi(c)∂),

λcab =

1

2(i(a)(e(∂cei(b) − ∂bei(c)) − i(c)e(∂bei(a) − ∂aei(b)) + i(b)e(∂aei(c) − ∂cei(a))).

and the compatibility conditions

∂mei(k) = ∂kei(m),

imply that the matrix λc becomes the zero matrix,

λc = 0.

3.3. The one-index Christoffel matrix of the second kind

The last result givesγm = hm = e∂me = γm,

γmab = γmab

or∂me = e−1γm = eσm,

σmab = σm

ab

where

γm =1

2(∂mg − ∂i(m)g + gi(m)∂)

σm ≡ e−1∂me = g−1γm

or

σm ≡ 1

2g−1(∂mg − ∂i(m)g + gi(m)∂),

where the elements of the one-index matrix σm are the Christoffel symbols Γamb of the second kind,

σmab =

1

2g−1aλ (∂mgλb − ∂λgmb + ∂bgλm) = Γa

mb = g−1aλ γ

mλb = g−1

aλ γbλm,

We call the matrix σm as the one-index Christoffel matrix of the second kind. The matrices γm andσm have the following property,

γmi(k) = γki(m)

σmi(k) = σki(m).

AUTH

OR

COPY


3.4. The first derivatives of the non-integrable but differentiable matrices e and e−1

Therefore the first derivatives of the non-integrable but differentiable matrices e and e−1 have thefollowing form:

∂me = e−1γm = eσm

∂me−1 = −e−1(∂me)e−1 = −g−1γme−1 = −σme−1

3.5. The second derivatives of the non-integrable but differentiable matrices e, and e−1

Using the first derivatives it is easy to obtain the second derivatives of the non-integrable but differen-tiable matrices e and e−1:

∂a∂be = e(∂aσb + σaσb), a = 1, . . . , n − 1; b = 2, . . . , n; b > a

∂a∂be−1 = (−∂aσb + σbσa)e−1

a = 1, . . . , n− 1; b = 2, . . . , n; b > a

Acting by the differential compatibility operator

[∂∂]ab = ∂a∂b − ∂b∂a

on the frame matrix e, we obtain

[∂∂]abe = eσab,

where the two-index matrix σab is:

σab ≡ ∂aσb − ∂bσ

a + σaσb − σbσa = e−1[∂∂]abe,

a = 1, . . . , n− 1; b = 2, . . . , n; b > a

σabmk = σ

abmk

The Riemann curvature tensor is identified as

Rmkab = ∂aΓ

mbk − ∂bΓ

mak + Γm

asΓsbk − Γm

bsΓsak

It is easy to see that the elements σabmk of the matrix σab are the elements of the Riemann curvature

tensor Rmkab,

σabmk = ∂aσ

bmk − ∂bσ

amk + σa

mλσbλk − σb

mλσaλk = ∂aΓ

mbk − ∂bΓ

mak + Γm

aλΓλbk − Γm

bλΓλak = Rm

kab,

or in the matrix form,

σab = g−1i(μ)i(λ)Rμλab = i(μ)i(λ)Rμλab,

where

σm = i(α)i(β)Γαmβ .

AUTH

OR

COPY


We introduce also a two-index matrix γab, where

γab = e[∂∂]abe = gσab,

γabmk = γabmk,

γabmk = gmλσabλk = gmλ(∂aΓ

λbk − ∂bΓ

λak + Γλ

aρΓρbk − Γλ

bρΓρak ) = gmλR

λkab = Rmkab .

In the last expression Rmkab is the Riemann tensor with all lower indices.From the compatibility conditions


we find

[∂∂]abei(c) = ∂a∂bei(c) − ∂b∂aei(c) = ∂a∂bei(c) − ∂b∂cei(a),

[∂∂]bcei(a) = ∂b∂cei(a) − ∂c∂bei(a) = ∂b∂cei(a)− ∂c∂aei(b),

[∂∂]caei(b) = ∂c∂aei(b) − ∂a∂cei(b) = ∂c∂aei(b)− ∂a∂bei(c),

Sabc ≡ [∂∂]abei(c) + [∂∂]bcei(a) + [∂∂]caei(b) = 0,

where Sabc is the sum of cyclic permutations of the indices a, b and c, which is vanishes due to thecompatibility conditions.

From the definition of the two-index matricesγab and σab

γab = e[∂∂]abe,

σab = e−1[∂∂]abe,

[∂∂]abe = e−1‘γab = eσab

we obtain

Sabc = e−1(γabi(c) + γbci(a) + γcai(b)) = e(σabi(c) + σbci(a) + σcai(b))

and the following properties of the two-index matrices γab and σab:

γabi(c) + γbci(a) + γcai(b) = 0,

σabi(c) + σbci(a) + σcai(b) = 0.

From the compatibility conditions


and from the integrability of the metric matrix g = ee

[∂∂]mn(ee) = 0 = ([∂∂]mne)e+ e[∂∂]mne = eγmne+ eγmne = σmng + gσmn,

we find that the matrices γab and gσab are antisymmetric matrices,

γab = −γab, gσab = −σabg.

AUTH

OR

COPY


Using the two previous properties of the matrix γab we will write the following expressions:

2γabmn = 2i(m)γabi(n) = −i(n)γabi(m)− i(m)γbai(n)

2γabmn = −i(n)(−γmai(b)− γbmi(a)) − i(m)(−γnbi(a)− γani(b))

2γabmn = −i(b)γmai(n)− i(a)γbmi(n)− i(a)γnbi(m)− i(b)γani(m)

2γabmn= i(b)γnmi(a)+ i(b)γani(m)+ i(a)γnbi(m)+ i(a)γmni(b)− i(a)γnbi(m)− i(b)γini(m)

2γabmn = i(b)γnmi(a) + i(a)γmni(b) = i(b)γnmi(a) + i(a)γmni(b) = 2i(a)γmni(b) = 2γmnab

The matrix γmn has an additional property: it is invariant under exchange of the upper pair of indiceswith the lower pair:

i(m)γabi(n) = i(a)γmni(b), γabmk = γmkab

Acting by the integrability operator on the matrix e and taking into a consideration that e0 is theintegrable function, we obtain

[∂∂]abe = e−1‘γab = eσab,

[∂∂]abe = ([∂∂]abΩ)e0,

Ωe0σab = ([∂∂]abΩ)e0.

The Riemann space time becomes flat if the two-index matrix γab (or the two-index matrix σab)becomes zero matrix.

From the two last equations it follows

eσabe−1 = ([∂∂]abΩ)Ω,

e0σabe−1

0 = Ω([∂∂]abΩ).

Therefore, the Riemannian spacetime becomes flat if and only if the orthogonal matrix Ω is integrable.The integrability of the matrix function e0 in the expression

e = Ωe0

implies

[∂∂]abΩ = Ωe0σabe−1

0

From the last equation it follows that the orthogonal matrix Ω (or e) is integrable if and only if thetwo-index matrix σab (or the two-index matrix γab) is a zero matrix. The integrability of the orthogonalmatrix Ω leads to the integrability of the frame matrix e and the vector Q.

4. The absolute Ricci, Riemann curvature and the absolute form of the Bianchi identities

4.1. The absolute compatibility operator matrix

In the Chapter 2 the absolute compatibility matrix operator [DD] and the coordinate compatibilitymatrix operator [∂∂]ab were introduced which are determined by the elements

[DD]mμ ≡ DmDμ −DμDm.

AUTH

OR

COPY


and

[∂∂]ab = ∂a∂b − ∂b∂a.

These two operators are connected by the expression

[DD] = e−1[∂∂]e−1 = e−1(i(α)i(β)[∂∂]αβ)e−1,

which is the transformation from the twice covariant compatibility matrix operator [∂∂] to the twicecovariant absolute compatibility matrix operator [DD].

4.2. The absolute form of the Ricci matrix

Now we introduce a new absolute matrix K, which is determined by the expression:

K ≡ ([DD]e)e−1 = −e([DD]e−1) .

Taking into the consideration that e0 is an integrable function, we obtain also

K = ([DD]Ω)Ω.

The twice covariant form of the matrix K is found from the matrix K,

K = eKe.

Then we obtain

K = eKe = e[DD]e = ee−1[∂∂]

↓e−1 e

↑= i(α)i(β)e−1[∂∂]αβe = i(α)i(β)σαβ

From the expressions for the elements of the matrix σab and the Riemann tensor Rmkab,

σabmk = ∂aσ

bmk − ∂bσ

amk + σa

msσbsk − σb

msσask =

∂aΓmbk − ∂bΓ

mak + Γm

asΓsbk − Γm

bsΓsak

Rmkab = ∂aΓ

mbk − ∂bΓ

mak + Γm

asΓsbk − Γm

bsΓsak

it follows,

σabmk = Rm

kab

Kmn = i(m)Ki(n) = i(m)i(α)i(β)σαβi(n) = σmββn = Rβ

nmβ

Kmn=σmββn =∂mσβ

βn−∂βσmβn+σm

βλσβλn−σβ

βλσmλn)=∂mΓβ

βn − ∂βΓβmn + Γβ

mλΓλφn − Γβ

βλΓλmn

The last expression coincides with the definition of the Ricci matrix. Therefore we can consider thematrix K as the absolute form of the Ricci matrix.Special notation. For the twice covariant matrix K special notation is used,

R ≡ K = i(α)i(β)σαβ = i(α)i(β)g−1γαβ

The matrix R is the twice covariant Ricci matrix and therefore the matrix K is the absolute form ofthe Ricci matrix.

AUTH

OR

COPY


4.3. The absolute Riemann curvature matrix

The absolute matrix Kmn = {Kmnij } is determined by

Kmn ≡ ([DD]mne)e−1 = −e([DD]mne

−1) .

It is easy to obtain the following equations

Kmn = (e−1i(α)i(β)e−1)mneσαβe−1,

Kμναβ = e−1

μme−1nν eαρσ

mnργ e−1

γβ ,

eaαebβ ecμedνKμναβ = gaρσ

cdρb = gaρR

ρbcd = Rabcd,

from which it follows that the matrix Kmn = {Kmnij } is the absolute curvature matrix or the absolute

matrix form of the Riemann-Christoffel tensor Rabcd.

4.4. Properties of the absolute Riemann curvature and Ricci matrices

From the integrability of the metric matrix g = ee it follows

[DD]mn(ee) = 0 = ([DD]mne)e+ e[DD]mne = eKmne+ eKmne,

and therefore the matrix Kmn is the antisymmetric matrix,

Kmn + Kmn = 0.

Using the integrability conditions

Dje−1i(k) = Dke

−1i(j), j, k = 1, . . . , n

implies other properties of the absolute Riemann curvature matrix K. Using the definition of K we canwrite

Kmni(k) =−e(Dm(Dne−1i(k)) −Dn(Dme−1i(k)))

=−e(Dm(Dne−1i(k)) −Dn(Dke

−1i(m))),

and then we obtain

Kmni(k) +Knki(m) +Kkmi(n) =−e(Dm(Dne−1i(k)) −Dn(Dke

−1i(m)))

−e(Dn(Dke−1i(m)) −Dk(Dme−1i(n)))

−e(Dk(Dme−1i(n))−Dm(Dne−1i(k))) = 0.

The last expression we can write in two forms,

Kmni(k) +Knki(m) +Kkmi(n) = 0,

i(k)Kmn + i(m)Knk + i(n)Kkm = 0.

The next property of matrices Kmn is found, using the previous property.

2Kmnrk = Kmn

rk +Knmkr = −Knr

mk −Krmnk −Kmk

nr −Kknmr = Knr

km +Krmkn +Kmk

rn +Kknrm

AUTH

OR

COPY


2Kmnrk = −Krk

nm −Kknrm +Krm

kn +Kmkrn +Kkn

rm −Krknm +Krm

kn +Kmkrn

2Kmnrk = −Krk

nm −Kmkrn −Kkr

mn +Kmkrn = 2Krk

mn

Hence we have

Kmnrk = Krk

mn.

The matrix γk depends only on g, and therefore satisfies the integrability conditions:

[DD]mnγk = 0 = [DD]mn(e∂ke) = ([DD]mne)∂ke+ e[DD]mn∂ke,

From the last equation it follows

[DD]mn∂ke = −e−1([DD]mne)∂ke = Kmn∂ke.

Using this expression, one obtains

[DD]mnDke= ([DD]mne−1ks )∂se+ e−1

ks [DD]mn∂se =

−Kmnkr e−1

rs ∂se+ e−1ks K

mn∂se = Kmnkr Dre+KmnDke,

and hence

[DD]mnDke = Kmnkr Dre+KmnDke.

Using the notation

Dmnk ≡ [DD]mnDk + [DD]nkDm + [DD]kmDn,

and the equality

[DD]mnDke = Kmnkr Dre+KmnDke

we obtain the following expressions

Dmnke = KmnDke+KnkDme+KkmDne+ (Kmnkr +Knk

mr +Kkmnr )Dre,

[DD]mnDke+ [DD]nkDme+ [DD]kmDne = Dmnke = KmnDke+KnkDme+KkmDne.

Then the following identity is written in two forms

Dmnk = Dmnk(1) = Dmnk(2),

where

Dmnk(1) ≡ [DD]kmDn + [DD]mnDk + [DD]nkDm,

Dmnk(2) ≡ Dk[DD]mn +Dm[DD]nk +Dn[DD]km.

For the first form

Dmnk(1)e = ([DD]mnDk + [DD]nkDm + [DD]kmDn)e = (KmnDk +KnkDm +KkmDn)e.

AUTH

OR

COPY


For the second form

Dmnk(2)e=Dk(Kmne) +Dm(Knke) +Dn(K

kme)

= (DkKmn +DmKnk +DnK

km)e+ (KnkDm +KmnDk +KkmDn)e.

From the equality

Dmnk(1)e = Dmnk(2)e,

it follows

DkKmn +DmKnk +DnK

km = 0,

or in the form for the matrix elements

DkKmnab +DmKnk

ab +DnKkmab = 0.

The last two equations present the absolute matrix form of the Bianchi identities. The matrices K andKmn are connected by the following contracting:

Kna = Kβnaβ = Knβ

βa .

The matrix element Kan can be obtained by means of the equality Kmnrk = Krk

mn,

Kan = Kβanβ = Knβ

βa = Kna.

Hence, the matrix K is the symmetric matrix,

K = K.

From the equation,

DkKmnab +DmKnk

ab +DnKkmab = 0,

with b = k = β and a = m = α it follows

DβKαnαβ +DαK

nβαβ +DnK

βmαβ = −DβKnβ −DαKnα +DnKαα = 0,

or

K↑D↓= DmKi(m) =

1

2D 〈K〉 .

4.5. Properties of the absolute Riemann curvature and Ricci matrices

We have obtained the following properties of the absolute Riemann curvature and Ricci matrices:1. Kmn + Kmn = 0

2. Kmnkr +Knk

mr +Kkmnr = 0

3. Kmnrk = Krk

mn.4. Kλn

aλ = Knλλa = Kna = Kan.

5. K = K.6. K

↑D↓= DλKi(λ) = 1

2D 〈K〉.7. DkK

mn +DmKnk +DnKkm = 0

AUTH

OR

COPY


5. The absolute velocity vector and the absolute field matrix

5.1. The absolute (non-integrable) and coordinate (integrable) velocity vectors

Using the notations: U for the absolute velocity-vector and u(x) for the coordinate velocity-vector,we can write the following definitions,

u(Q) ≡ dQdτ ,

U ≡ u(Q) ,

u(x) ≡ dxdτ ,

There is the following connection between these two velocity vectors,

U = eu(x).

From the expressions for the infinitesimal invariant interval

dτ2 = dQdQ .

and

dτ2 = dxgdx .

it follows

UU = 1,

and

u(x)gu(x) = 1.

5.2. The absolute non-integrable but differentiable field matrix

The full differential dU of the absolute velocity vector U is written as

dU = (U↑D↓)dQ,

Then it is possible to write

dUdτ = (U

↑D↓)dQdτ = (U

↑D↓)U = PU ,

where the absolute matrix P is,

P ≡ U↑D↓= ePxe

−1 .

Px ≡ u(x)∂ + σαu(x)i(α) =→P = e−1Pe

We call the absolute matrix P as the absolute mathematical field matrix or shortly the absolute fieldmatrix.

AUTH

OR

COPY


5.3. Two types of absolute field matrices: The absolute symmetric and the absolute antisymmetric fieldmatrices

The absolute matrix P can be written as the sum of two matrices

U↑D↓= P = Pa + Ps,

where the antisymmetric absolute field matrix Pa is

Pa ≡ 1

2(U↑D↓−DU) = −Pa,

and the symmetric absolute field matrix Ps is

Ps ≡ 1

2(U↑D↓+DU) = Ps.

5.4. The coordinate field matrices

In the n - dimensional coordinate space {x} we can write the coordinate field matrices: - the twicecontravariant matrix P ,

P ≡ e−1P e−1 ,

the twice covariant matrix P ,

P ≡ eP e

where

P = g(σαu(x)i(α) + u(x)∂) ,

the covariant-contravariant matrix P→

P→≡ eP e−1 = gP = Pg−1

and the contravariant-covariant matrix→P ,

→P ≡ e−1Pe = Pg = g−1P .

For the coordinate antisymmetric P a and symmetric P s field matrices we find the following expres-sions,

P a ≡1

2(P − P ),

P s ≡1

2(P + P ),

AUTH

OR

COPY


5.5. The absolute equation of motion

From the equations

UU = 1,

UPaU = 0,

d(UU)

dτ= U(Ps + Pa)U + U(Ps + Pa)U = 2UPsU = 0,

it follows that the symmetric absolute field matrix must satisfy the condition

PsU = 0 .

Therefore

dUdτ = (U

↑D↓)U = (Pa + Ps)U = PaU .

We call the last equation as the absolute equation of motion.

5.6. The coordinate equation of motion

Substituting into the absolute equation of motion the following expressions,dU

dτ=

de

dτu+ e

du

dτ,

de

dτ= eσμuμ,

PaU = e−1Pau,

we obtain

du

dτ= −σμuμu+ g−1Pau

We call the last equation as the coordinate equation of motion, where

σμ = i(α)i(β)Γαμβ , σμ

βα = Γβμα.

The scalar coordinate equation of motion

duβ(x)

dτ= g−1

βλP aλμuμ(x)− σμ

βαuμ(x)uα(x) = g−1βλP aλμ

uμ(x)− Γβμαuμ(x)uα(x).

for Pa = 0 becomes

duβ(x)dτ = −σμ

βαuμ(x)uα(x) = −Γβμαuμ(x)uα(x) ,

which is the differential equation of geodesic, the well known from the tensor classical theory of generalrelativity.

AUTH

OR

COPY


5.7. The 4-dimensional space: The coordinate equation of motion of a charged particle in the presenceof both gravitational and electromagnetic fields

The coordinate equation of motion of a charged particle in the presence of both gravitational andelectromagnetic fields is written in the form [3]:

duβ(x)

dτ=

e

mc2F βμ uμ(x)− Γβ

μαuμ(x)uα(x),

where F βμ is the tensor of an electromagnetic field. If we compare the last equation with the scalar

coordinate equation of motion

duβ(x)

dτ= g−1

βλP aλμuμ(x)− σμ

βαuμ(x)uα(x) = g−1βλP aλμ

uμ(x)− Γβμαuμ(x)uα(x),

we will see that the antisymmetric matrix→P a,

→P a = g−1P a ,

is the matrix of an electromagnetic field.If the antisymmetric field matrix Pa is zero and there is only the symmetric field matrix Ps, the

equation of motion becomes the following one:

duβ(x)

dτ= −σμ

βαuμ(x)uα(x) = −Γβμαuμ(x)uα(x),

which in the four-dimensional space is the equation of motion in a gravitational field. Therefore, thesymmetric field matrix Ps in the four-dimensional space is the matrix of a gravitational field and thefield matrix P can be considered as an electromagnetic-gravitational field matrix.

6. The field matrix identities and derivation of the equations of the field

6.1. The absolute (non-integrable in the non-flat Riemannian space) field matrix

The absolute equation of motion of a mathematical point is

dUdτ = (U

↑D↓)dQdτ = (U

↑D↓)U = PU ,

where the absolute matrix P is,

P ≡ U↑D↓

.

We call the absolute matrix P as the absolute field matrix. We know that,

u(x)gu(x) = UU = 1.

AUTH

OR

COPY

https://www.researchgate.net/publication/270920140_The_Classical_Theory_of_Field?el=1_x_8&enrichId=rgreq-1a448e863c07f98a5b188a8094e498f0-XXX&enrichSource=Y292ZXJQYWdlOzI1ODQwNDIwNztBUzoyMzA0OTk1MDc1MDMxMDRAMTQzMTk2Njc3MzMyMQ==


U = eu(x), u(x) ≡ dx

dτ, D = e−1∂.

de

dτ= eσαuα(x).

The matrix P can be written as the sum of two matrices

U↑D↓= P = Pa + Ps,

where the antisymmetric absolute field matrix Pa is

Pa ≡ 1

2(U↑D↓−DU) = −Pa,

and the symmetric absolute field matrix Ps is

Ps ≡ 1

2(U↑D↓+DU) = Ps.

In the 4-dimensional spacetime the geometrized coordinate time is x4,

x4 = ct.

Here c is the constant velocity of light but it can also be the function of t, where t is the coordinatetime.

The matrix P must satisfy the condition

d(UU)

dτ= U(Ps + Pa)U + U(Ps + Pa)U = 2UPsU = 0.

That means the symmetric absolute field matrix Ps must satisfy the condition

PsU = 0 .

or the coordinate equation

P su(x) = 0 .

6.2. The equations of motion

The coordinate form of the absolute equation of motion is

du(x)dτ = g−1P au(x)− σαu(x)uα(x) .

The scalar variant of the last coordinate form isduβ(x)

dτ= g−1

βλP aλμuμ(x)− Γβ

μαuμ(x)uα(x).

For Pa = 0 the scalar coordinate equation of motion becomes the well known differential equation ofgeodesic.

The coordinate vector-velocity u(x) is an integrable function. Therefore

[DD]u(x) = 0.

The expression for [DD]U can be written in the form,

[DD]U = D〈UD〉 − (DU)D = D〈P 〉 − (P )D .

AUTH

OR

COPY


6.3. The first field matrix equation

We know that in the case of 4-dimensional Riemannian spacetime the antisymmetric matrix Pa is theabsolute form of the electromagnetic field which satisfies the absolute first group of Maxwell equation

D〈Pa〉 − (Pa)D = PaD = ρaU .

We can suppose, that there is the first group of the generalized Maxwell equation

D〈P 〉 − (P )D = ρU .

where the equation

PaD = ρaU .

is the first group of the Maxwell equations for the antisymmetric field matrix and the equation

D〈Ps〉 − PsD = ρsU .

is the first group of the generalized Maxwell equations for the symmetric field matrix. Here ρa is theinvariant density for the antisymmetric field and ρs is the invariant density for the symmetric field, and

ρ ≡ ρa + ρs.

6.4. The second matrix field equation

The second absolute field matrix identity links the Riemann curvature matrix Kmn with the absolutefield matrix P . The expression for [DD]mnU can be written in the form,

[DD]mnU = KmnU .

The second absolute field matrix identity can be written also in the form:

KmnUk = DmDnUk −DnDmUk.

Using the absolute identityi(k)Kmn + i(m)Knk + i(n)Kkm = 0,

and the second absolute field matrix identity we obtain the second group of the absolute field equations,

DmPakn +DnPamk +DkPanm = 0 .

or the second group of the coordinate field equations,∂ρP aμν + ∂μP aνρ + ∂νP aρμ = 0,

We use the following notation for the antisymmetric field matrix Pa for a four dimensional spacetime(n = 4, x4 = ct):

Pa ≡ P (k, l), P (k, l) ≡

⎛⎜⎜⎝

0 k3 −k2 l1−k3 0 k1 l2k2 −k1 0 l3−l1 −l2 −l3 0

⎞⎟⎟⎠ ,

and for the corresponding dual antisymmetric matrix Pa,

Pa ≡ P (l, k), P (l, k) =

⎛⎜⎜⎝

0 l3 −l2 k1−l3 0 l1 k2l2 −l1 0 k3−k1 −k2 −k3 0

⎞⎟⎟⎠ ,

AUTH

OR

COPY


6.5. The electromagnetic-gravitational field

The equation

∂ρP aμν + ∂μP aνρ + ∂νP aρμ = 0

in four-dimensional space has the same form as the second group of Maxwell equations for the twicecovariant electromagnetic field matrix, which can be written in the matrix form for the dual covariantfield matrix,

∧Pa ∂ = ∂αP (l, k)i(α) = 0.

That means we can consider the absolute field matrix Pa in 4 – dimensional metric space as theabsolute electromagnetic field matrix and therefore the equation

DmPakn +DnPamk +DkPanm = 0,

is the absolute form of the second group of the Maxwell equations, which can be written also in theform,

Pa↑D↓= P

↑(l, k)D

↓= 0 .

If in the 4-dimensional spacetime the antisymmetric absolute field matrix Pa is the absolute elec-tromagnetic field matrix, we can suppose that the symmetric absolute field matrix Ps is the absolutegravitational field matrix, satisfying the absolute equations,

D〈Ps〉 − PsD = ρsU .

PsU = 0 .

Therefore in the 4-dimensional spacetime we can consider the matrix P as the absolute field matrixof electromagnetic-gravitational field which satisfies the generalized Maxwell equations,

D〈P 〉 − (P )D = ρU .

DmPakn +DnPamk +DkPanm = 0 ,

and the equation

PsU = 0 ,

where ρ,

ρ ≡ ρa + ρs,

can be considered as the sum of invariant charge and mass densities.A. Einstein writes [2]: The only clue which can be drawn from experience is the vague perception that

something like Maxwell’s electromagnetic field has to be contained within the total field”.

AUTH

OR

COPY

https://www.researchgate.net/publication/266712706_The_meaning_of_relativity_5th_ed?el=1_x_8&enrichId=rgreq-1a448e863c07f98a5b188a8094e498f0-XXX&enrichSource=Y292ZXJQYWdlOzI1ODQwNDIwNztBUzoyMzA0OTk1MDc1MDMxMDRAMTQzMTk2Njc3MzMyMQ==


6.6. The matrix metric equation

The expression for [DD]U can be written also in another form,

[DD]U = ([DD]e)u(x) = ([DD]e)e−1U = KU .

Therefore we obtain the first absolute field matrix identity,

KU = D 〈P 〉 − (P )D .

and the matrix metric equation

KU = ρU .

g−1Ru(x) = ρu(x) .

Einstein’s field equations have the form:

R− 1

2〈g−1R〉g + Λg = T.

In his last book A. Einstein writes [2]: The left-hand side of this equation depends only on the sym-metrical tensor, gik, which represents the metric properties of space as well as the gravitational field. Theright-hand side of this equation is a phenomenological description of all the sources of the gravitationalfield. Tik represents the energy which generates the gravitational field, but it itself of non-gravitationalcharacter, as for example the energy of the electromagnetic field, of the density of ponderable matter etc.To represent this tensor, Tik, concepts are taken from pre-relativity physics which only a posteriori havebeen adapted to the principle of general relativity.

Such a dualistic treatment of the total field could not be avoided during the first phase of the develop-ment of relativity. It was, however, evident that this represented only a provisional treatment of the totalproblem.”

In our theory the metric equation is only another form of the field matrix equation.

6.7. The equation of continuity

The invariant equation of continuity for the electromagnetic-gravitational field is

D(ρU) = 0 .

For the gravitational field the invariant rest-mass density satisfies the equation of continuity written indifferent forms,

D(ρsU) = dρs

dτ + ρs〈Ps〉 = ∂(√

−|g|ρsu(x))√−|g| = ∂(|L|ρsu(x))

|L| == 12d〈g−1R〉

dτ + 〈g−1Rg−1Ps〉 = 0 .

For the electromagnetic field the field matrix Pa is an antisymmetric matrix. Therefore

〈Pa〉 = 0 ,

D(PaD) = 0 ,

and the equation of continuity is an expression of the law of charge conservation

D(ρU) = D(PD)) = dρa

dτ = 0 .

AUTH

OR

COPY



7. The general spherically symmetric solution of the metric matrix equation for the symmetricalmatrix field in the four-dimensional metric space

Let us apply our matrix theory for finding the spherically symmetric solution of the metric matrixequation for the symmetrical matrix field in the four-dimensional metric space. This is the case of thegravitational field. We use the “geometrized units”, where all quantities which in ordinary units havedimension expressible in terms of length, time and mass, are given the dimension of a power of length.All scalars, vector and matrix elements, which we use, are in the geometrized units.

7.1. The “rectilinear” and “spherical” coordinate systems

We use the “rectilinear” coordinate system with the four-dimensional vector y and the corresponding“spherical” coordinate system with the four-dimensional vector x, where

y1 = x3 cos x1, y2 = x3 sinx1 sinx2,y3 = x3 sinx1 cos x2, y4 = x4 = ct

.

Here c is the speed of light, t is time,

x3 =√

y21 + y22 + y23 = r, x1 = ϑ, x2 = ϕ.

7.2. The general spherically symmetric metric matrix

In the spherical coordinate system the metric matrix can be presented in the following form,

g = {{−h2(x3)x23, 0, 0, 0}, {0,−h2(x3)x23 sin2 x1, 0, 0},{0, 0,−h2(x3),−x3h(x3)f(x3)},{0, 0,−x3h(x3)f(x3), 1

h6(x3)− x23f

2(x3)}},

Determinant of this metric matrix g is the same as the determinant of the metric matrix of the flatspace time:

Det(g) = −x43 sin2 x1.

We will write the metric matrix in the short form,

g = {{g11, 0, 0, 0}, {0, g22, 0, 0}, {0, 0, g33 , g34}, {0, 0, g34, g44}},

where

g11 = −h2(x3)x23, g22 = −h2(x3)x23 sin2 x1, g33 = −h2(x3),g34 = −x3h(x3)f(x3), g44 =

1

h6(x3)− x23f

2(x3)

AUTH

OR

COPY


7.3. The 4-velocities

The solution of the equations of motion (geodesic equations for the symmetrical matrix field) givesthe expressions for 4-velocities,

u1(x) = −√

c21 − c22 cot2 x1

g11

u2(x) = − c2g22

u3(x) = c3

√g44g33

(c21 + c22x23

− g33

)+ c24g

233

u4(x) = −c4 + g34u3(x)

g44

Here

c3 = ±1.7.4. General nonsingular spherically symmetric solution of the gravitational equations

Using the short notations,

h ≡ h(x3), f ≡ f(x3),

we will find the four eigenvalues μ1, μ2, μ3, μ4 of the square matrix g−1R,

μ1 =1

x3h4

(x33h

8(f ′)2 − 9x3(h′)2 + x3f

2h7(3h+ 4x3h′) + x23fh

7(2f ′(3h+ 2x3h′)

+x3hf′′) + 3h(2h′ + x3h

′′))

)

μ2 =1

x3h4

(x33h

8(f ′)2 − 13x3(h′)2 + x23fh

7(2f ′(3h + 2x3h′) + x3hf

′′)+h(−2h′ + x3h

′′) + x3f2h6(3h2 + 4x23(h

′)2 + 2x3h(6h′ + x3h

′′)

)

μ3 = μ4 =1

h4(3(h′)2 + 2x3fh

7f ′(h+ x3h′)− hh′′ + f2h6(3h2 + 3x23(h

′)2

+x3h(8h′ + x3h

′′)))

The 4-velocity vector u(x) will be the eigenvector for the equal eigenvalues

μ1 = μ2 = μ3 = μ4 = ρ.

For the difference μ1 − μ2 we obtain the following expression,

μ1 − μ2 = −2(−1 + x23f2h6)(2x3(h

′)2 + h(4h′ + x3h′′))

x3h4.

It is possible to check, that the function

h =

(1−

(c5x3

)3)1/3

AUTH

OR

COPY


satisfies the differential equation

μ1 − μ2 = 0.

Using the above expression for the function h we find the difference μ2 − μ3:

μ2 − μ3 =1

x123

(1−

(c5x3

)3)8/3(−10x43c35(x33 + c35) + (x33 − c35)

2(−c65f2 + x23(x33 − c35)

2(f ′)2

+x3f(2(2x63 − 3x33c

35 + c65)f

′ + x3(x33 − c35)

2f ′′)))

The solution of the differential equation

μ2 − μ3 = 0

is the following function f :

f =c8(c

35(2x

33 − c35) +

2c63 (x33 − c35)

5/3 + c7(x33 − c35)

8/3)1/2

x3(x33 − c35)

.

In the last expression

c8 = ±1For the obtained functions h and f we obtain the equal eigenvectors,

μ1 = μ2 = μ3 = μ4 = 3c7 = ρ.

Therefore

c7 =ρ

3,

where ρ has the dimension of density and we have the following expression for the Ricci matrix:

R = ρg.

Now we can write the following expressions:

h =

(1−

(c5x3

)3)1/3

f =c8(c

35(2x

33 − c35) +

2c63 (x33 − c35)

5/3 + c7(x33 − c35)

8/3)1/2

x3(x33 − c35)

g44 = 1− 2c63x3h

− c7x23

The constants of integration can be determined from the motion. The dimension of c6 is a length,which is proportional to the gravitational radius of the rest mass M of the gravitating body and is writtenas

c63

= rm,

AUTH

OR

COPY


where rm is the gravitational radius of the rest mass M of the gravitating body,

rm ≡ GgM

c2,

and Gg is the gravitational constant. The constant c5 is proportional to the gravitational radius of the restmass m of the body, moving in the gravitational field of the rest mass M .

We have found a rigorous nonsingular spherically symmetric solution of the gravitational equationswhich may be applied to the investigation of the motion of planets around the Sun. At this case the massM is the rest mass of the Sun. The second constant c7 can be determined from the measurements of theprecession of perihelia. For the motion of Mercury we obtain c7 = ρ

3 ≈ 1.3 ∗ 10−33km−2 if the majoraxis of Mercury’s orbit precesses at a rate of 43.1 arcsecs every 100 years.

7.5. The equation of the plane orbital motion

For the plane orbital motion, where

y1 = const = y10 =π

2, c1 = 0

the radius x3 as a function of x2 for the general spherically symmetric metric matrix is determined bythe differential equation

d2k

dx22=

1

3kc22(−3k2c22 − 3d(1 − k3c35)

2/3c7 + 3k5c22(dc6 + c35(5 − 4dc7))

+4k8c22c35(−2dc6 + 3c35(−1 + dc7) + 4k6c35(1− k3c35)

2/3(−2dc6 + 3c35(−1 + c24 + dc7))

+k3(1− k3c35)2/3((dc6 − 3c35(−4 + 4c24 + 3dc7))), k ≡ 1

x3, d =

(1

k3− c35

)2/3

For the particular case

c5 = 0, h = 1, c7 = 0,c63

= rm,

we obtain

f = c8

√2rmx3

x3,

g = {{g11, 0, 0, 0}, {0, g22 , 0, 0}, {0, 0, g33 , g34}, {0, 0, g34 , g44}}g11 = −x23, g22 = −x23 sin2 x1, g33 = −1,

g34 = −c8√

2rmx3

, g44 = 1− 2rmx3

AUTH

OR

COPY


8. The general spherically symmetric solution for c5 = 0

8.1. The metric matrix and 4-velocities

Let us consider the particular case of the general spherically symmetric solution for c5 = 0, h = 1.For this case of the weak gravitational force we obtain the following expressions:

g = {{−x23, 0, 0, 0}, {0,−x23(sin x1)2, 0, 0},{0, 0,−1,−c8

√2rmx3

+ c7x23

},

{0, 0,−c8

√2rmx3

+ c7x23, 1−2rmx3− c7x

23

}c8 = ±1,R = 3c7g

u1(x) =

√c21 − c22 cot

2 x1x23

,

u2(x) =c2

(x3 sinx1)2,

u3(x) = c3

√c24 −

(1− 2rm

x3− c7x23

)(c21 + c22x23

+ 1

), c3 = ±1′

u4(x) = −c4 − c8u3(x)

√2rmx3

+ c7x23

1− 2rmx3− c7x

23

.

For the particular case of the zero density ρ,

c7 = 0,

we obtain

f = c8

√2rmx3

x3,

g = {{g11, 0, 0, 0}, {0, g22 , 0, 0}, {0, 0, g33 , g34}, {0, 0, g34 , g44}}g11 = −x23, g22 = −x23 sin2 x1, g33 = −1,

g34 = −c8√

2rmx3

, g44 = 1− 2rmx3

8.2. The general form of spherically symmetric solution for c5 = 0 (weak solution) with one arbitraryfunction

It is possible to show that the general form of spherically symmetric solution for c5 = 0 (weaksolution) is

g = {{−x23, 0, 0, 0}, {0,−x23 sin2 x1, 0, 0}, {0, 0, g33(x3), c8√

1 + g33(x3)g44(x3)},{0, 0, c8

√1 + g33(x3)g44(x3), g44(x3)}}.

AUTH

OR

COPY


where g33(x3) is an arbitrary function,

g44 = 1− 2rmx3− ρ

3x23.

This form implies that the radius vector x3 is defined in such a way that the circumference of a circlewith center at the origin of coordinates is equal to 2πx3 (the element of arc of a circle in the planex1 = π/2 is equal to ds = x3dx2) [3]. The coefficient c8 = 1 for x4 > 0 and c8 = −1 for x4 < 0. Atthis case the term g34dx3dx4 + g43dx4dx3) will be invariant under the time reversal.

For this weak metric matrix we have

R = ρg.

8.3. The Schwarzschild solution

The Schwarzschild diagonal metric matrix gs is the particular case of the general weak metric matrix,where the function g33(x3) is taken in the form:

g33(x3) = − 1

g44(x3)= − 1

1− 2rmx3− ρ

3x23

.

For this function g33(x3) and for ρ = 0 we obtain the Schwarzschild metric matrix gS ,

g=gS≡{{−x23, 0, 0, 0}, {0,−(x3 sinx1)2, 0, 0},

{0, 0,− 1

1 − 2rmx3

, 0

},

{0, 0, 0, 1 − 2rm

x3

}},

which has a singularity at the Schwarzschild radius x3 = 2rm. S. Hawking comments this singularity [4]:“However, this is just caused by a bad choice of coordinates. One can choose other coordinates in whichthe metric is regular there.”

8.4. The 4-velocities for the general weak metric matrix with ρ = 0

For the general weak metric with ρ = 0,

g={{−x23, 0, 0, 0}, {0,−x23 sin2 x1, 0, 0},{0, 0,−1,−c8

√2rmx3

,

{0, 0,−c8

√2rmx3

, 1− 2rmx3

}}we find

u1(x) = −√

c21 − c22 cot2 x1

x23,

u2(x) =c2

y23 sin2 x1

,

u3(x) = c3

√c24 −

(1− 2rm

x3

)(c21 + c22x23

+ 1

), c3 = ±1,

u4(x) = −c4 − c8u3(x)

√2rmx3

1− 2rmx3

.

Then dimensionless velocities become

β1(x3) =u1(x)x3u4(x)

, β2(x3) =u2(x)x3u4(x)

, β3(x3) =u3(x)

u4(x).

AUTH

OR

COPY



8.5. The plane orbital motion [1]

For the plane orbital motion with ρ = 0, where

x1 = const =π

2, c1 = 0, u1(x) = 0,

the constants are expressed through the perihelion distance p and the aphelion distance a, taking intoconsideration that u3(x) = 0 at x3 = p and at x3 = a,

c2 = −ap√2rm

f2, c4 =

f3f2

,

f2 =√

ap(a+ p)− 2rm(a2 + ap+ p2),

f3 =√

(a+ p)(a− 2rm)(p − 2rm) =√

ap(a+ p)h(a)h(p).

Then we have

u2(x) =

√2rmap

f2x23

,

u3(x) =

√2rmf1

f2x3/23

,

u4(x) =f3x3

f2(x3 − 2rm)+

2rmf1(x3 − 2rm)f2x3

.

with

f1 =√

(a− x3)(x3 − p)(a(px3 − 2rm(x3 + p))− 2rmpx3).

For the Schwarzschild model

u4(x) =f3x3

f2(x3 − 2rm)

This is the only difference between our model and the Schwarzschild model. The extreme dimension-less transverse velocities are the same for the both metrics and are written in the form:

β2max = −β2(a) = −a

p

√2rm(p − 2rm)

(a+ p)(a− 2rm),

β2min = −β2(p) = −p

a

√2rm(a− 2rm)

(a+ p)(p − 2rm).

The radius x3 as a function of x2 for the both models is determined by the same differential equation

d2k

dx22= −k + 3k2rm +

rmc22

, k =1

x3.

Let x21 be the position of perihelion and x22 is the position of perihelion after one period of revolution.Using a method of successive approximation, we find the difference Δϕ, which gives the displacementof perihelion after one period of revolution,

Δϕ = x22 − x21 − 2π =3rmf4f5

,

AUTH

OR

COPY


Table 1Relativistic precession (arc sec) of planets (observed and using our formulae)

Precession per revolution formula Precession per century observed Precession per century formulaMercury 0.103517 43.1 ± 0.5 42.9531Venus 0.0530579 8.4 ± 4.8 8.6273Earth 0.0383884 5.0 ± 1.2 3.83884Mars 0.0254106 1.35091Jupiter 0.00739156 0.0623129Saturn 0.0040177 0.0136392Uranus 0.00200285 0.00238403Neptune 0.00127736 0.000775147Pluto 0.00104024 0.000419995

Table 2Calculated extreme orbital velocities (km/s) of planets (using our formulae) and the appropriate NASA data

vmin Nasa data vmin Formula vmax Nasa data vmax FormulaMercury 38.86 38.8568 58.98 58.9779Venus 34.79 34.7850 35.26 35.2575Earth 29.29 29.2903 30.29 30.2879Mars 21.97 21.9708 26.50 26.5017Jupiter 12.44 12.4327 13.72 13.7103Saturn 9.09 9.0927 10.18 10.1815Uranus 6.49 6.49358 7.11 7.11496Neptune 5.37 5.37276 5.50 5.49512Pluto 3.71 3.70514 6.10 6.10229

where

f4 = a2p2(a2 − p2) + 4(a2 + ap+ p2)rm(ap2 − rm(a2 + ap+ p2)),

f5 = a2p2(a− p)(ap− 2rm(2a+ p)).

The Table 1 shows the precession of planets as the results of calculations using the expression

Δϕ = x22 − x21 − 2π =3rmf4f5

In the Table 2 we show the calculated extreme velocities of planets, using the formulae

β2max = −β2(a) = −a

p

√2rm(p − 2rm)

(a+ p)(a− 2rm),

β2min = −β2(p) = −p

a

√2rm(a− 2rm)

(a+ p)(p − 2rm).

and the appropriate NASA data.

8.6. The case of purely radial motion

For the case of purely radial motion

β10 = 0, β20 = 0, y1 =π

2.

AUTH

OR

COPY


In the Schwarzschild geometry the dimensionless radial velocity becomes

β3 = ±y3 − 2rmc4y3

√c24 − 1 +

2rmy3

.

The radial velocity reaches an extreme value β3m,

|β3m| = 2c243√3,

at the radius y3m, where

y3m =2rm

1− 4c449

,

c4 =

(1− 2rm

y30

)√√√√√ 1− 2rmy30(

1− 2rmy30

)2 − β230

.

If the particle begins accelerating at large radius (y3 →∞) with zero initial velocity β30 (c4 → 1), itwill attain a maximum value β3max,

|β3max| = 2

3√3= 0.39.

For y3 → 2rm, β3 → 0.We have obtained the known result that for the Schwarzschild geometry, in the region 2rm < y3 <∞,

the velocity of the particle undergoes variations unlike the monotonic increase predicted by Newtoniangravitational theory.

In our geometry the dimensionless radial velocity becomes

β3r(y3) =

(1− 2rm

y3

)√α24r −

(1− 2rm

y3

)√

2rmy3

(α24r −

(1− 2rm

y3

))± α4r

,

with

α4r =1− 2rm

y30− β30

√2rmy30√

1− 2rmy30− 2β30

√2rmy30− β2

30

.

For the particle which begins accelerating with zero initial velocity β30 we find for the negative veloc-ity

α4r0 =

√1− 2rm

y30,

AUTH

OR

COPY


β3r0(y3) =

(1− 2rm

y3

)√1− 1− 2rm

y3

1− 2rmy30√

2rmy3

(1− 1− 2rm

y3

1− 2rmy30

)− 1

.

For the particle which begins accelerating at large radius (y30 →∞) with zero initial velocity β30 wefind for the negative velocity

α4r0 → 1, β3r0(y3)→ −√

2rmy3

.

For y3 → 2rm the value of β3r0(2rm) aspires to −1.We see that in our geometry the velocity of the particle undergoes the monotonic increase predicted

by the Newtonian gravitational theory.

9. The radius of mass and the radius of charge of charged mass

For the gravitational field of mass the radius of mass (the mass in the units of length) is

rm ≡ GgM

c2,

where Gg is the gravitational constant and M is the central mass.For the electromagnetic field of the charged mass instead the radius of mass we need to take the radius

of charge rc in the units of length,

rc =q2

4πε0mcc2,

where ε0 is electric constant, q is elementary electric charge, mc is the mass of the charge.For example, for the electromagnetic field in the hydrogen atom the radius of proton charge rc in the

units of length is

rc =q2

4πε0mpc2,

where mp is rest mass of proton. Using the radius of the atom ra,

ra =ε0�

2

πmeq2,

where 2π� is Planck constant, me is electron rest mass, we can obtain the relation rc/ra,rcra∼= 2.9 × 10−8.

For the comparison we find the relations rm/p and rm/a for the Mercury – the nearest planet to theSun,

rmp∼= 3.2× 10−8,

rma∼= 2.1× 10−8.

AUTH

OR

COPY


Here p – the perihelion, a – the aphelion of the Mercury. These data tell about some analogy betweenthe electromagnetic and gravitational fields.

Let us find the relation between the radius of proton charge rcp to the radius of proton mass rmp, where

rcp ≡ q2

4πε0mpc2, rmp ≡ Gmp

c2.

We get

rcprmp

=q2

4πε0Gm2p

= 1.23587 ∗ 1036.

The obtained number is the relation between the electromagnetic and gravitational forces.

10. Friedmann-lobachevsky model

10.1. The metric and the Ricci matrices

The Friedmann-Lobachevsky model [5], is starting from the assumption that the expressions for themetric matrix is written in the form,

g = f2(s)G,

where

s ≡√xGx,

ds

dτ=

xGu(x)

s.

10.2. The Christoffel matrices of the second kind

For this metric matrix we find the Christoffel matrices of the second kind,

σ1 =f ′(s)sf(s)

⎛⎜⎜⎝−x1 −x2 −x3 x4x2 −x1 0 0x3 0 −x1 0x4 0 0 −x1

⎞⎟⎟⎠ ,

σ2 =f ′(s)sf(s)

⎛⎜⎜⎝−x2 x1 0 0−x1 −x2 −x3 x40 x3 −x2 00 x4 0 −x2

⎞⎟⎟⎠ ,

σ3 =f ′(s)sf(s)

⎛⎜⎜⎝−x3 0 x1 00 −x3 x2 0−x1 −x2 −x3 x40 0 x4 −x3

⎞⎟⎟⎠ ,

σ4 =f ′(s)sf(s)

⎛⎜⎜⎝

x4 0 0 −x10 x4 0 −x20 0 x4 −x3−x1 −x2 −x3 x4

⎞⎟⎟⎠ ,

AUTH

OR

COPY

https://www.researchgate.net/publication/44537353_The_theory_of_space_time_and_gravitation_by_V_Fock?el=1_x_8&enrichId=rgreq-1a448e863c07f98a5b188a8094e498f0-XXX&enrichSource=Y292ZXJQYWdlOzI1ODQwNDIwNztBUzoyMzA0OTk1MDc1MDMxMDRAMTQzMTk2Njc3MzMyMQ==


10.3. The equations of motion

It is possible to verify that geodesic equation

du(x)dτ = −σαu(x)uα(x) ,

for the metric matrix

g = f2(s)G

is satisfied, if we take u(x) in the form

u(x) =dx

dτ=

x

sf(s).

For this uλ(x) we obtain

ds

dτ=

1

f(s)

du(x)

dτ=

d

dτ

(x

sf(s)

)= −xf ′(s)

sf3(s).

du(x)

dτ= −σαu(x)uα(x) = −xf ′(s)

sf3(s)

Using the equations

u(x) =dx

dτ=

x

sf(s)

ds

dτ=

1

f(s).

we obtain

dx

s=

x

s.

The solution of the last differential equation is

xλ = aλs,

where aλ is constant and

a4 =√

1 + a21 + a22 + a23.

The dimensionless 3-velocities are introduced,

βα ≡ dxαdx4

=uα(x)

u4(x)=

aαa4

, α = 1, 23; .

AUTH

OR

COPY


β ≡√

a21 + a22 + a23a4

=√

β21 + β2

2 + β23 .

Therefore we have

a4 =1√

1− β2.

xλ =βλ√1− β2

s, λ = 1, 23;

x4 =s√

1− β2.

Since s2 > 0 we must have β2 < 1.

10.4. The expression for the Ricci matrix

Now we can obtain the expressions for the Ricci matrix,

R =1

(sf(s))2(h1 − h2)gxxg + h2g = (h1 − h2)gu(x)u(x)g + h2g

K = h1UU + h2(I − UU),

where

h1 ≡ 3(−s(f ′(s))2 + f(s)(f ′(s) + sf ′′(s)))sf4(s)

,

h2 ≡ s(f ′(s))2 + 5f(s)f ′(s) + sf(s)f ′′(s)))sf4(s)

.

From the equations

KU = ρU,

UKU = ρ

it follows

h1 = ρ.

The coordinate metric equation is

g−1Ru(x) = ρu(x).

It is easy to verify the following expressions,

Det(g−1R) = |g−1R| = h1h32,

T rK = 〈K〉 = Tr(g−1R) = 〈g−1R〉 = h1 + 3h2,

Det(g−1R− μI) = |g−1R− μI| = (μ− h1)(μ − h2)3.

AUTH

OR

COPY


10.5. The field matrix

Using the expression,

u(x) =dx

dτ=

x

sf(s)

we find the expressions for the field matrix,

P = h(I − UU),

P = h(g − gu(x)u(x)g),

where

h ≡ f(s) + sf ′(s)sf2(s)

= −s d

ds

(1

sf(s)

).

T rP = 〈P 〉 = ⟨g−1P⟩= 3h,

10.6. Solving the matrix metric equation

Solving the equation∣∣g−1R− μI∣∣ = (μ− h1)(μ − h2)

3 = 0,

for the square matrix

g−1R = (h1 − h2)u(x)u(x)g + h2I,

we find four roots or four eigenvalues,

μ1 = h1, μ2 = μ3 = μ4 = h2.

The matrix metric equation

(g−1R− μI)u(x) = 0,

for the first root μ = μ1 = h1 is satisfied

(g−1Ru(x)− μ1u(x) = (h1 − h2)u(x)u(x)gu(x) + h2u(x)− μ1u(x)

= (h1 − h2)u(x) + h2u(x)− h1u(x) = 0

because

R = (h1 − h2)gu(x)u(x)g + h2g

and

u(x)gu(x) = 1.

For the second root we obtain

(g−1Ru(x)− μ2u(x) = (h1 − h2)u(x) + h2u(x)− μ2u(x) = (h1 − h2)u(x).

Let us consider at first the case of the second root μ = μ2.

AUTH

OR

COPY


10.7. The first variant: μ = μ2. The case of a maximally uniform space

The equation

(h1 − h2)u(x) = 0

will be satisfied, if

h1 − h2 =2(−2s(f ′(s))2 + f(s)(−f ′(s) + sf ′′(s)))

sf4(s)= 0.

The solution of the last differential equation determines the function f(s) and the density ρ:

f(s) =1

a− bs2,

ρ = h2 = h1 = 12ab.

where a and b are constants and therefore the density ρ is also constant. Putting a = 1 we get

b =ρ

12,

so that

f(s) =1

1− ρ12s

2.

The twice covariant Ricci matrix in this variant has the following form,

R = ρg.

This is the case of a maximally uniform space [5]. In the considered case the constant of curvature k,

k =1

12〈K〉 = 1

12〈g−1R〉 = ρ

3.

Using the obtained results we can write

f(s) =1

1− k4s

2=

1

1− ρ12s

2,

g =1

1− k4s

2G =

1

1− ρ12s

2G.

We have received the same result, as the similar result in [5], which was obtained using the Einsteinequations of gravitation.

AUTH

OR

COPY




10.8. The second variant: μ = μ1. The case of a Big Bang

The matrix metric equation

(g−1R− μI)u(x) = 0,

for the first root μ = μ1 = h1 is satisfied

ds

dτ=

1

f(s)

From the equation of continuity

D(ρsU) = dρs

dτ + ρs〈Ps〉 = 0

and from the equationsds

dτ=

1

f(s),

P = h(I − UU),

T rP = 〈P 〉 = 〈g−1P 〉 = 3h,

h ≡ f(s) + sf ′(s)sf2(s)

= −s d

ds

(1

sf(s)

),

it follows,

dρ(s)

dτ=

dρ(s)

ds

ds

dτ=

dρ(s)

f(s)ds= −ρ(s)〈P 〉 = −3ρ(s)h.

Solving the equation of continuity, we find

ρ(s) =b

(sf(s))3.

or

f(s) =b1/3

sρ(s)1/3,

where b is constant. Substituting the last expression into the formulae for μ1, μ2 and h we obtain

μ1 = −s(−s(ρ′(s))2 + ρ(s)(ρ′(s) + sρ′′(s))b2/3ρ4/3(s)

= ρ(s),

μ2 =−18(ρ(s))2 + 5s2(ρ′(s))2 − 3sρ(s)(ρ′(s) + sρ′′(s))

9b2/3ρ4/3(s),

h = − sρ′(s)3b1/3(ρ(s))2/3

.

AUTH

OR

COPY


The general solution of the equation μ1 = ρ(s) is

ρ(s) =8d6

27b2

(ssm

)d(1 +

(ssm

)d/3)6 = ρm64(

ssm

)d(1 +

(ssm

)d/3)6 .

Here ρm, sm and d are constants. Therefore, the expression for f(s) will be

f(s) =

3b

(1 +

(ssm

)d/3)2

2d2s(

ssm

)d/3 ,

The density ρ(s) has maximum ρ = ρm at s = sm, where

sm = x4m√

1− β2.

ρ(s) = ρm64(

ssm

)d(1 +

(ssm

)d/3)6

ρm =d6

216b2.

Therefore, the expression for f(s) will be

f(s) =

d

(1 +

(ssm

)d/3)2

4s√6ρm

(ssm

)d/3 .

The general form of the metric matrix at this case becomes

g =

d2(1 +

(ssm

)d/3)4

96s2ρm

(ssm

)2d/3 G,

10.9. The mathematical model of the Big Bang

In the Friedmann-Lobachevsky space with the metric matrix g,

g = f2(s)G,

f(s} =(1 +

a

s

)2,

xλ = βλx4,

s = x4√

1− β2.

AUTH

OR

COPY


Fig. 1. The invariant mass density as a function of time in the Friedmann-Lobachevsky space. The continuous line is our

functionρ(

ssm

)ρm

=64

(s

sm

)d

(1+

(s

sm

)d/3)6 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 [6].

a = sm = x4m√

1− β2.

each mass moves with a constant velocity proportional to its coordinates and therefore in this space alldistances increase proportionately with time.

The general mathematical model of the Big Bang is described by the expressions

g =

d2(1 +

(ssm

)d/3)4

96s2ρm

(ssm

)2d/3 G,

and

ρ(

ssm

)ρm

=64(

ssm

)d(1 +

(ssm

)d/3)6

Figure 1 shows two curves. The continuous line is our functionρ(

s

sm

)

ρmfor d = 5.4. The dotted line

is the cosmic microwave background spectrum from COBE, where ρ is the intensity and s is wavelengths/centimetre [6].

11. The flat n-dimensional space

11.1. The definition of the flat n-dimensional space

In the tensor classical theory of general relativity the space becomes flat if all elements of the Riemanncurvature tensor Rm

kba,m, k, b, a = 1, . . . , n become zero. In the matrix language the n-dimensional

AUTH

OR

COPY


space becomes flat if all elements of the two-index coordinate cotravariant–covariant Riemann curvaturematrix σab become zero.

11.2. The coordinate frame matrix e0 for the flat n-dimensional sapace

Is it possible to find such coordinate frame matrix

e0 = e0flat

as a function of all elements of the coordinate n-vector x, for which the Riemann curvature matrix σab

is zero? Yes, and this matrix is

e0flat ≡ I + Fa(xk)(I − I(k))xi(k)

where Fa(xk) is an antisymmetric n×n matrix function of xk. For this coordinate frame matrix we find

σkflat = e−1

0flat (Fa(xk)e0flat + ∂ke0flat )

σmflat = e−1

0flat∂me0flat ), m �= k

Taking into the consideration the equation for the orthogonal matrix Ω,

Ω∂aΩ = (e0σa − ∂ae0)e

−10 , a = 1, . . . , n

for the flat space we find

Ωflat∂mΩflat = 0, m �= k

Ωflat∂kΩflat = Fa(xk), k = 1, . . . , n

Therefore the orthogonal matrix function Ωflat depends only on the variable xk and is determined bythe differential equation

dΩflat

dxk= ΩflatFa(xk)

11.3. The absolute frame matrix e for the flat n-dimensional space

The absolute frame matrix e for the flat n-dimensional space takes the form:

eflat = Ωflate0flat

or

eflat = Ωflat(I + Fa(xk)(I − I(k))xi(k))

11.4. The coordinate metric matrix g for the flat n-dimensional space

The coordinate metric matrix g for the flat n-dimensional space is denoted by gflat , where

gflat ≡ e0flate0flat

AUTH

OR

COPY


11.5. The connection between the full differentials dQ and dx for the flat n-dimensional space

The connection between the full differentials dQ and dx

dQ = edx

for the flat n–dimensional space becomes

dQflat = Ωflat(I + Fa(xk)(I − I(k))xi(k))dx

Therefore the quadratic form

dτ2 = dxgflatdx

is reducible to

dτ2 = dQflatdQflat

Here Q can be presented as the sum of two vector functions,

Qflat = M +W,

where

M ≡∫ xk

0Ωflatdxk

W ≡ Ωflat(I − I(k))x

dM = ΩflatI(k)dx

dW = dΩflat (I − I(k))x +Ωflat(I − I(k))dx

= ΩflatFa(xk)(I − I(k))xi(k)dx +Ωflat(I − I(k))dx

From the last two equations it follows the expression for dQflat ,

dQflat = dM + dW = Ωflat(I + Fa(xk)(I − I(k))xi(k))dx,

which proves that the solution of the differential equation

dQflat = Ωflat(I + Fa(xk)(I − I(k))xi(k))dx

is

Qflat = M +W =

∫ xk

0Ωflatdxk +Ωflat(I − I(k))x

where the orthogonal matrix function is defined by the differential equation

dΩflat

dxk= ΩflatFa(xk).

AUTH

OR

COPY


12. The orthogonal matrix function as a function of the antisymmetric matrix

From the antisymmetric matrix function it is possible to obtain the orthogonal matrix Ω,

Ω = Ω(P (k, l)) = a1I + a2P (l, k) + a3S(P (k, l)) + a4P (k, l),

or

Ω = Ω(P (k, l)) = a11I + a2P (l, k) + a3P2(k, l) + a4P (k, l).

The matrix Ω has six functions determined by the 3-dimensional vectors k and l and four functionswhich form the coefficients ak. These coefficients are the functions of the invariants kl and kk+ ll. Thecoefficients have the following form:

a1 ≡ 1

2(cos b+ cos a),

a11 ≡ ω21 cos b− ω2

2 cos a

ω21 − ω2

2

,

a2 ≡ kl

ω21 − ω2

2

(sin b

ω2− sin a

ω1

),

a3 ≡ cos b− cos a

ω21 − ω2

2

,

ω21 ≡

1

2

(kk + ll

)±√α

ω22 ≡

1

2

(kk + ll

)∓√α

α ≡ 1

4(kk + ll)2 − (kl)2

ω21 − ω2

2 = ±2√αω21 + ω2

2 = kk + ll

ω21ω

22 = (kl)2.

Here a and b are functions of x. Some other notations will be used also,

b2 ≡ ω1 sin b− ω2 sin a,

b3 ≡ cos b− cos a,

b4 ≡ ω1 sin a− ω2 sin b,

B1 ≡ a1I, B2 ≡ b2ω21 − ω2

2

P (l, k),

and

B3 ≡ b3ω21 − ω2

2

S(P (k, l)), B4 ≡ b4ω21 − ω2

2

P (k, l).

AUTH

OR

COPY


It is possible to show, that

ΩΩ = I,

that means the matrix Ω is the orthogonal matrix, which is determined by the antisymmetrical matrixP (k, l). The orthogonal matrix Ω(P (k, l)) has the following commutative property,

Ω(P (k, l))P (k, l) = P (k, l)Ω(P (k, l)).

For

k = const, l = const, a ≡ ω1s, b ≡ ω2s,

one obtains

dΩ(P (k, l))

ds= P (k, l)Ω(P (k, l)).

13. The absolute G-vectors, G-orthogonal and G-antisymmetric matrix functions in4-dimensional space-time

13.1. The z-space-time

In order to avoid the use of imaginaries the constant Minkowski metric matrix G and the correspondingmatrix e(G) are introduced, where

G ≡

⎛⎜⎜⎝−1 0 0 00 −1 0 00 0 −1 00 0 0 1

⎞⎟⎟⎠ ,

e(G)e(G) ≡ G

Let e(G) has a simplest form

e(G) ≡

⎛⎜⎜⎝

i 0 0 00 i 0 00 0 i 00 0 0 1

⎞⎟⎟⎠ .

We introduce the absolute G-vectors and G-matrices, replacing the absolute vectors Q, U , D, theframe matrix e0, the absolute matrix P , the Ω-absolute matrix P0 and the orthogonal matrix Ω,

z ≡ e−1(G)Q,

u(z) ≡ e−1(G)U,

D(z) ≡ e(G)D

L ≡ e−1(G)e0

AUTH

OR

COPY


Pz ≡ e−1(G)Pe(G)

Λ ≡ e−1(G)Ωe(G)

Φ ≡ e−1(G)P0e(G) = e−1(G)Ω−1PΩe(G)

L ≡ e−1(G)e0

Pz ≡ e−1(G)Pe(G)

Λ ≡ e−1(G)Ωe(G)

Φ ≡ e−1(G)P0e(G) = e−1(G)Ω−1PΩe(G)

The matrix Λ is a G-antisymmetric matrix, where

ΛGΛ = G.

The matrix Φ is the Λ-absolute matrix Φ,

Φ = G−1ΛGPzΛ.

The infinitesimal interval for the z-space is written in the form

δτ2 = dzGdz,

where G is the flat metric and z is an integrable vector function with the complete differential dz,

dz = e−1(G)dQ = ΛLdx.

13.2. The absolute G-antisymmetric matrix

The absolute G-antisymmetric matrix function Φ satisfies two equations of G- and G−1-antisymmetry:

GΦ + ΦG = 0,

and

ΦG−1 +G−1Φ = 0.

The following notations are used:

κ ≡ {{κ1}, {κ2}, {κ3}}, λ ≡ {{λ1}, {λ2}, {λ3}},κλ ≡ κ1λ1 + κ2λ2 + κ3λ3, κκ ≡ κ21 + κ22 + κ23, λλ ≡ λ2

1 + λ22 + λ2

3,

Φ ≡ Φ(κ, λ), Φ(κ, λ) ≡

⎛⎜⎜⎝

0 κ3 −κ2 λ1

−κ3 0 κ1 λ2

κ2 −κ1 0 λ3

λ1 λ2 λ3 0

⎞⎟⎟⎠ ,

∧Φ ≡ Φ(λ,−κ), Φ(λ,−κ) =

⎛⎜⎜⎝

0 λ3 −λ2 −κ1−λ3 0 λ1 −κ2λ2 −λ1 0 −κ3−κ1 −κ2 −κ3 0

⎞⎟⎟⎠ .

AUTH

OR

COPY


Φ(κ, λ) = P (κ, λ)G = −G−1P (κ, λ)

∧Φ(κ, λ) = −

∧P (κ, λ)G

Φ(λ,−κ) = −Φ−1(κ, λ)(κλ).

and

Φ(λ,−κ) = −Φ−1(κ, λ)(κλ).

13.3. The G-orthogonal matrix function of four variables [1]

Using the new introduced matrix functions Λ, Φ (κ,λ), Φ (−λ,κ), and L, one obtains

g = LGL,

Λ = c1I + c2Φ(λ,−κ) + c3S(Φ(κ, λ)) + c4Φ(κ, λ),

or

Λ = c11I + c2Φ(λ,−κ) + c3Φ2(κ, λ) + c4Φ(κ, λ).

Here the following notations are used

c1 =1

2(cosh b+ cos a),

c11 =w21 cosh b+ w2

2 cos a

w21 + w2

2

,

c2 = − kl

w21 + w2

2

(sinh b

w2− sin a

w1

),

c3 =cosh b− cos a

w21 + w2

2

,

c4 =w1 sin a+ w2 sinh b

w21 + w2

2

,

w21 =

1

2

(√q + κκ− λλ

),

w2 =κλ

w1,

q = (κκ− λλ)2 − 4(κλ)2,

w21 + w2

2 =√q,

w21 − w2

2 = κκ− λλ,

For

κ = const, λ = const, a = w1s, b = w2s,

AUTH

OR

COPY


it follows

dΛ(Φ(κ, λ))

ds= Φ(κ, λ)Λ(Φ(κ, λ)),

where

Φ(κ, λ) = Φ = G−1L−1PL−1.

The G-orthogonal matrix Λ has the following commutative property,

Λ(Φ(κ, λ))Φ(κ, λ) = Φ(κ, λ)Λ(Φ(κ, λ)).

Instead of the metric matrix e0 the new frame metric matrix L will be used in the metric equation. Allmetric matrices and dz are written below, using L and G-orthogonal matrix Λ,

e0 = e(G)L,

e = e(G)ΛL,

g = LGL = LΛGΛL,

dz = ΛLdx,

g = LGL = LΛGΛL,

dz = ΛLdx.

The frame matrix equation in this case is written in the form:

ΛG∂αΛ = G(Lσα−∂αL)L−1 =

1

2L−1((∂αL)GL)L−1− LG∂αL+gi(α)∂− i(α)g)L−1 = GΦ.

14. Summary

A new matrix theory of mathematical field in the n-dimensional metric space is developed. The motionof mathematical points in this field is studied. Two spaces are considered: the n-dimensional Riemannianspace and the n-dimensional space of non-integrable vectors and matrices. We call the last space as theabsolute space. The absolute mathematical field matrix is introduced and two forms of the matrix fieldequations are derived. Applications to the four-dimensional Riemannian spacetime are considered. It isshown that in this case the absolute field matrix is the absolute electromagnetic-gravitational field matrix.The symmetric part of this matrix is the absolute gravitational field matrix. The antisymmetric part is theabsolute electromagnetic field matrix. The differential equations for these two field matrices and for themetric matrix have been obtained. The partial case of these equations presents the Maxwell equationsfor the electromagnetic field. Four general solutions for the static spherically symmetric gravitationalfield are found. Two these solutions have no an apparent singularity at the Schwarzschild radius andthey correspond to the constant invariant density which is not zero (the Schwarzschild metric has zerodensity). For the first solution, which contains the parameter, which can be interpreted as one mass, theformulae for the minimum and maximum orbital velocity and for the perihelion precession of planetsthrough perihelion and aphelion distances are given. In our calculations of extreme velocities of planets

AUTH

OR

COPY


all first four digits coincide with the NASA data. The second solution is more general solution and itcontains two parameters, which can be interpreted as two masses.

The obtained matrix field equations were solved also for a maximally uniform space. We obtainedthe expression for the density of particles which looks like the expression describing the mathematicalmodel of the Big Bang, where the density at the beginning of time is zero and after that it is growingvery fast until very big maximum and after this maximum it is slowly decreasing. Our expression iscompared with the experimental cosmic microwave background spectrum from COBE, where ρ is theintensity and s is wave lengths/centimeter [6].

Acknowledgment

This research was supported from NSF grant #0960222. I also thank Dr. G.A. Glass for his continuedsupport of my research.

References

[1] A.D. Dymnikov and G.A. Glass, An introduction to the matrix classical theory of field, New Formalism, Equations andSolutions, St Peterburg: Dmitrii Bulanin (2005), 64.

[2] A. Einstein, The meaning of relativity, Princeton University Press, Princeton, New Jersey (1953).[3] L.D. Landau and E.M. Lifshitz, The classical theory of fieds, Butterworth-Heinemann Ltd Oxford London (1994).[4] S. Hawking and R. Penrose, The nature of space and time, Princeton University Press Princeton New Jersey (1996).[5] V.A. Fok, The theory of space, Time and gravitation, Macmillan Co/Pergamon Press, New York (1964).[6] D.J. Finsen, E.S. Cheng, J.M. Gales, J.C. Mather, R.A. Shafer and E.L. Wright, The cosmic microwave background

spectrum from the full COBE/FIRAS Data Set, Astrophys J 473 (1996), 576.

AUTH

OR

COPY




https://www.researchgate.net/publication/1812435_The_Cosmic_Microwave_Background_Spectrum_from_the_Full_COBEFIRAS_Data_Set?el=1_x_8&enrichId=rgreq-1a448e863c07f98a5b188a8094e498f0-XXX&enrichSource=Y292ZXJQYWdlOzI1ODQwNDIwNztBUzoyMzA0OTk1MDc1MDMxMDRAMTQzMTk2Njc3MzMyMQ==



sign data derivative recovery

Documents