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European Journal of Applied Sciences – Vol. 11, No. 1
Publication Date: February 25, 2023
DOI:10.14738/aivp.111.14043. Blanovsky, A. (2023). The Lorentz Theory Revisited. European Journal of Applied Sciences, Vol - 11(1). 527-532.
Services for Science and Education – United Kingdom
The Lorentz Theory Revisited
Anatoly Blanovsky
Westside Environmental Technology Company, Los Angeles, USA
Abstract: Long before quantum mechanics, Lorentz suggested that some
disturbances, like waves, travel with particles through a certain medium -
motionless ether. It is shown that these waves are described by the Klein-Gordon
equation and dispersion relation w2=c2k2+wc
2. In hydromechanics, they are known
as non-propagating waves and a constant wc is called the cutoff frequency. A
quantum object is considered as a vibrating particle or material body with a rest
mass m moving in resonance with wave characterized by the cutoff or Compton
frequency wc=mc2/ .
Keywords: Non-propagating waves, Maxwell's equations, Matter/Anti-matter cosmology
INTRODUCTION
We derive here the general expressions relating the space and time coordinates of an arbitrary
point in two frames of reference in relative motion. Let x1, x2, x3 be coordinates of this point in
time t in the frame K. In the moving frame K', position of the same point is defined by the
coordinates x'i in time t'. For the points fixed in frame K' we have dx'i =0 and dxi =Vi dt. The
infinitesimal changes of xi and t in the fix point of frame K' are equal and
. Since the same point moves with the velocity in frame K
(1)
(2)
Assuming that two frames are equivalent in homogeneous and isotropic medium, we require
the covariance of these equations in frames K and K'. As inverse transformations for equal- direction coordinate axes differ only by the velocity sign, we have and
.
Introducing for convenience a new function we obtain
!
( )
dt
t
t r t dt ¢ ¶ ¢
¶ ¢ ¢ = , !
( )
dt
t
x r t dx i
i ¢ ¶ ¢
¶ ¢ ¢ = , !
V (r,t) ! !
dt (V gradt )dt
t
t dt + × ¢ ¶
¶ ¢ ¢ = !
dt V gradx dt
t
x dx i
i
i + ( × ¢) ¶
¶ ¢ ¢ = !
t
t r t
t
t r t
¶ ¢
¶ ¢ ¢ = ¶
¶ ¢( , ) ( , ) ! !
t
x r t
t
x r t i i
¶ ¢
¶ ¢ ¢ = - ¶
¶ ¢( , ) ( , ) ! !
2
2
2
) 1 (
) (
( )
t
t
t
V t
V
¶
¶ ¢ -
¶
¶ ¢
h = -
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European Journal of Applied Sciences (EJAS) Vol. 11, Issue 1, February-2023
(3)
, (4)
where B and Di are arbitrary vectors. If observed point is moving in the frame K with a velocity
v and consequently in K' with the velocity v', then dxi =vi dt and dx'i =v'i dt'.
We define in (3, 4) arbitrary vectors so that components of the vectors dxi and dx'i normal to
the velocity V are equal and grad t doesn't change at the normal to the velocity V displacement.
In this case, vectors B=0 and Di =[ni V]/V2.
Here ni is unit vector and (5)
. (6)
If a new frame K'' is moving with a velocity v' in frame K', then . Putting in
this formula dt' and v', we have in frame K .
But on the other hand in this frame . These two formulas are identical if h(V)=
h(v)= h(v')=constant. As this constant has dimensions of square of velocity, we mark it as a c2.
From demand that dt' is the total derivative, we have
and .
[ ]
( ) ( ) 1 2 VB
V V V
V gradt !! !
+
-
¢ =
h h
[ ]
( ) 1
2 2
i
i
i V VD
V V V
V gradx ! ! ! +
-
=
h
÷
÷
ø
ö
ç
ç
è
æ -
-
¢ = ( )
( ) 1
1
2 V
drV dt
V
V
dt
h
h
! !
V
V
drV V V dt dr
V
drV
V
V
dr ! ! ! ! ! ! ! ! ! 2 2 2
( ) 1
1
+ - ÷
÷
ø
ö
ç
ç
è
æ - ×
-
¢ =
h
( ) 1 2
v
v
dt dt
¢ ¢ -
¢¢ ¢ =
h
( ) 1 ( ) 1
( ) 1
2 2
2
v
v
dt
v
v
V
v
dt
h h
h
-
¢¢ ×
¢ ¢ -
¢ -
=
( ) 1 2
v
v
dt dt
h -
¢¢ =
0
1 2
2 =
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
- c
V
curl V
!
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
-
=
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
-
- ¶
¶
2
2
2
2 2 1
1
1
c
V
grad
c
c V
V
t
!
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529
Blanovsky, A. (2023). The Lorentz Theory Revisited. European Journal of Applied Sciences, Vol - 11(1). 527-532.
URL: http://dx.doi.org/10.14738/aivp.111.14043
With condition we have , where l is an arbitrary function.
It is convenient to express the Lorentz transformation in the two-dimensional coordinates x=x- ct and h=x+ct
and , (7)
where b=V/c is a dimensionless constant (Mach number) and q=tanh-1b. We have that the
Lorentz transformation expands one coordinate while contracting the other to preserve their
product or a wave equation in a moving reference system [1].
CLASSICAL FIELD THEORY
Harmonic oscillators occupy an important place in classical physics, including construction of
the Lorentz metrics representation and understanding the role of the phase. The Lorentz
transformation relating the space and time coordinates of two reference systems, K (xi, t) and
K' (xi', t'), in relative motion can be obtained as partial differential equations.
.
(8)
Here, vector is the velocity at a point fixed in reference system K', and c is the constant.
The Lorentz transformation keeps in moving frames a wave equation invariant instead of time
in Galilean transformation. The alteration of the time scale t' preserves in moving frames a wave
equation.
The function t' is a surface of constant phase and satisfies the eikonal or Hamilton-Jacobi
equation of mechanics in dimensionless form
. (9)
Derivatives of the function t' are a wave vector and frequency that are connected by a
dispersion relation w2=c2k2+wc
2. In hydromechanics, they are known as non-propagating
waves. Their group velocity approaches zero and waves are not propagating if their frequency
is below the cutoff frequency wc. The group and phase velocity are related by vu=c2, in the
infinite k limit v=u=c and the group velocity maximum is c.
curlV = 0 ! (V grad V ) r t V
t
V ! ! ! ! !
= - × ) = l( , ) ¶
¶
q x
b
b x = e
+
- ¢ = 1
1 q h
b
b h - = -
+ ¢ = e
1
1
2 2
2
1 ( ) ()
1
t drV dt dt V gradt dt dt
t V c
c
¶ ¢ ¢ ¢ = + × = - ¶ -
! ! !
2 2 2
2
1 ( )
1
drV drV dr V Vdt dr V
V V V
c
¢ = - + -
-
! ! ! ! !! ! ! !
Vrt (,)
! !
22 2 2 2
123
( ) ( ) ( ) ( ) 10 ttt t
c
xx x t
é ù ¶¶¶ ¶ ¢¢¢ ¢ ê ú + + - + =
¶¶ ¶ ¶ ë û