ISSUE 2007 VOLUME 2 - The World of Mathematical Equations

Volume 2 PROGRESS IN PHYSICS April, 2007
however their geometrical sense is not clear.
Thus the anisotropic field can only be a mixed vortical
gravitational field bearing both the “electric” and the “magnetic”
components. A strictly “electric” or “magnetic” vortical
gravitational field is always spatially isotropic.
Taking the above into account, we arrive at the necessary
and sufficient conditions for the existence of standing waves
of the gravitational inertial force:
1. A vortical gravitational field of the strictly “magnetic”
kind is the medium for standing waves of the gravitational
inertial force;
2. Standing waves of the gravitational inertial force are
permitted only in a non-stationary rotating space.
As soon as one of the conditions ceases, the acting gravitational
inertial force changes: the standing waves of the
force transform into traveling waves.
4 The field equations of a vortical gravitational field
It is known from the theory of fields that the field equations
of a field of a four-dimensional vector-potential A α is a
system consisting of 10 equations in 10 unknowns:
• Lorentz’s condition ∇σA σ = 0 states that the fourdimensional
potential A α remains unchanged;
• the continuity equation ∇σ jσ = 0 states that the fieldinducing
sources (“charges” and “currents”) can not
be destroyed but merely re-distributed in the space;
• two groups (∇σF ασ = 4π
c jα and ∇σF ∗ασ = 0) of the
Maxwell-like equations, where the 1st group determines
the “charge” and the “current” as the components
of the four-dimensional current vector jα of the field.
This system completely determines a vector field A α and
its sources in a pseudo-Riemannian space. We shall deduce
the field equations for a vortical gravitational field as a field
To deduce the Maxwell-like equations for a vortical gravitational
field, we collect together the chr.inv.-projections
of the field tensor Fαβ and the field pseudotensor F ∗αβ . Expressing
the necessary projections with the tensor of the rate
of the space deformation D ik to eliminate the free h ik terms,
we obtain
E i = 1
c hik ∗∂Fk ∂t
H ik = 2h im h kn ∗ ∂Amn
∗ i 1 ∂F 2
= +
c ∂t c Fk D ik , (39)
∂t =
∗ ik ∂A
= 2
∂t + 4 � A i∙
∙nD kn − A k∙
∙mD im� ,
H ∗i = ε imn ∗ ∂Amn
∂t
(40)
∗ ∗i ∂Ω
= 2
∂t + 2Ω∗iD , (41)
E ∗ik = − 1
c εikm ∗∂Fm . (42)
∂t
After some algebra, we obtain the chr.inv.-Maxwell-like
equations for a vortical gravitational field
∗ 2 i
1 ∂ F
c ∂xi ∗
2 ∂
+
∂t c ∂xi �
FkD ik� + 1
�
∗∂F
�
i
ik
+2FkD Δ
c ∂t j
ji− − 2
c Aik
�
∗∂Aik ∂t +Ai∙ ∙nD kn
�
= 4πρ
∗ 2 ik
∂ A
2
∂xk 1
−
∂t c2 ∗ 2 i ∗
∂ F ∂
+4
∂t2 ∂xk � i∙
A∙nD kn −A k∙
∙mD im� +
�
+2 Δ j
�
1
jk− Fk
c2 � ∗ ik
∂A
∂t +2�A i∙
∙nD kn −A k∙
∙mD im��
−
− 2
c2 ∗
∂ � ik
FkD
∂t
� − 1
c2 �
∗∂F i
∂t +2FkDik
�
D = 4π
c ji
⎫
⎪⎬
(43)
⎪⎭
⎫
∗ 2 ∗i
∂ Ω
∂xi∂t +
∗
∂
∂xi � � ∗i 1
Ω D +
c2 Ω∗m ∗ ∂Fm
+
ε ikm ∗ ∂ 2 Fm
∂xk �
+ εikm Δ
∂t j
jk
� ∗∂Ω ∗i
∗ ∗i
∂Ω
+ 4D
∂t +2
∂t + Ω∗i D
� ∗
1 ∂Fm
− Fk
c2 ∂t +2
∂t +
�
Δ j
ji
∗ ∂ 2 Ω ∗i
Group I,
= 0 ⎪⎬
Group II.
∂t2 +
�
∗∂D
∂t +D2
�
Ω ∗i = 0 ⎪⎭
of the four-dimensional potential F α = −2c2a∙α σ∙bσ .
Writing the divergence ∇σF σ σ
∂F =
∂xσ + Γσ σμF μ chr.inv.-form [2, 3]
in the
∇σF σ = 1
� ∗∂ϕ
c ∂t +ϕD
� ∗ i ∂q
+
∂xi +qi ∗∂ln √ h
∂xi 1
−
c2 Fiq i (37)
where ∗ ∂ ln √ h
∂xi = Δ j
ji and ∗ ∂q i
∂xi + qiΔ j
ji =∗∇i qi , we obtain
the chr.inv.-Lorentz condition in a vortical gravitational field
∗ i ∂F
∂xi + F i Δ j 1
ji −
c2 FiF i The chr.inv.-continuity equation ∇σj
= 0 . (38)
σ = 0 for a vortical
gravitational field follows from the 1st group of the Maxwelllike
equations, and is
∗ 2
∂
∂xi∂x k
�
∗∂A
�
ik
−
∂t
1
c2 �
∗∂Aik ∂t +Ai∙ ∙nD kn
�� �
∗
∂Aik
AikD+ −
∂t
− 1
c2 �
∗∂2 ik
A
∂t2 +
∗
∂�
i∙
A∙nD ∂t
nk��
Aik+ 1
2c2 �
∗∂F i
∂t +2FkDik
�
×
�∗∂Δj ∗
ji D ∂D
× + Fi−
∂t c2 ∂xi �
∗ 2
∂
+2
∂xi∂x k
� i∙
A∙nD kn −A k∙
∙mD im� +
�
∗∂Aik +
∂t +2�A i∙
∙nD kn −A k∙
∙mD im��� ∗
∂
∂xi �
Δ j
�
1
jk− Fk +
c2 �
+ Δ j
��
1
ji − Fi Δ
c2 l lk − 1
�
Fk
c2 �
(45)
= 0 .
(44)
To see a simpler sense of the obtained field equations, we
take the field equations in a homogeneous space (Δi km = 0)
6 D. Rabounski. The Theory of Vortical Gravitational Fields