Copyright (C) V. Skorobogatov, 2006
The Light in the 4D-Aether Model.
V.Skorobogatov
It is widely acknowledged that the Lorentz transformation is an
extension of the Galilean one related to the case of big velocities
when the theory of relativity ought to apply. Here we try to show that
there is another possibility leading to correct Lorentz transformation.
It seems to be produced by the simple rotation of the coordinate system
on the angle depentent from the velocity and therefore has no appliance
to the transformation between two reference frames moving with respect
each other. An explanation of the result of Michelson-Morley experiment
by using the Galilean transformation applied to 4D-aether model is based
upon some quite convincing assumption about velocity of photon. Also an
interpretation of known wave-particle duality is given.
1. The Lorentz transformation
For the beginning let us imagine that there is a body moving along the x-axis of
the fixed reference frame on some distance from the observer which resides at
the origin of the frame in the point A (Fig.1).
We assume that at the initial moment the body is at nearest distance from the
observer in the point B on the y-axes. The light signal emitting in that moment
will reach the body in the point C after some time interval t elapsed. For that
time the light passed a distance equals ct while the body moved at the distance
vt, where c is the light velocity that put constant here and v is the velocity
of body. If the latter would be equal zero the light signal will reach the body
for the less time interval t'. So we can take the angle CAB as the measure of
the velocity. Denoting that angle as α the following definition of the
velocity is arising
As it follows from this expression, the velocity of the body cannot exceed the
velocity of light. From the consideration of the triangle ABC it follows that
the distance ct' is equal ct cos α, and, after reducing c and using Eq.(1) to
express the angle α through the v, the time duration
One may treat the last equation as the time dilation like in the theory of
relativity. In that case we must prescribe time duration t' to the moving
reference frame and give it the next notion. It is the time needed for the
light signal to penetrate from the point B to the point A in the moving frame
and hence from the point of view of the unmoving observer located at the point
A it can erroneously consider as the same time as in his unmoving frame in the
case when the velocity of the body is vanished. If so, there is a "time delay'
caused by the movement.
However, we will get another interpretation of the picture. When the body
displaced from point B to C it looks like a rotation of the coordinate system
on the angle α. It may seem so to the observer at the point A. When he sees the
light signal emitted from the point B, he may suppose that the body is rotating
and therefore he can conclude that the coordinate system associated with the
body turns relatively his system.
Now it is easy to determine the distance along x-axes in both coordinate
systems. If to denote any point on the axe by x, the correspondent point lying
on the path of the body considered marked by letter E on the Fig.1 will have
the coordinate x' in the rotated coordinate system. The relation between these
coordinates is
|
x = x'/cos α + ct' tan α
|
(3) |
Here an expression ct' as the coordinate along y-axes is put down because
we measure the distance along that direction by means of light.
Respectively, the distance along y'-axes can be wrote as
|
ct = ct'/cos α+ x' tan α.
|
(4) |
Here the first term is nothing else than Eq.(2) multiplied by c and the second
one is an additional distance needed for light to surmount the length x'.
Substituting v from Eq.1 into Eqs. 3 and 4 we get formulas of Lorentz
transformation:
|
x = (x'+vt') / √(1 - (v/c)2),
|
(5) |
|
t = (t'+x'v/c2) / √(1 - (v/c)2)
|
(6) |
Hereby it was shown that the Lorentz transformation can be represented as a
rotation in usual space rather then in Minkovsky one. Instead of so called
interval s of the theory of relativity determined as
|
s = √((ct)2 - x2) = √((ct')2 - x'2),
|
(7) |
which may receive imaginary values, here it is a real distance keeping constant
value under rotation
|
d = √((ct')2 + x2) = √((ct)2 + x'2)
|
(8) |
It is the length from point A to point E on Fig.1. As it seen from the last
relations constancy of s is caused by constancy of d.
The following conclusions may be done from the analysis of derivation proposed
here. We know exactly that the body is not really spinning around the observer.
But we show that the Lorentz transfomation arises as a result of the simple
rotation in 3D space, and therefore there is no need to use artificial 4D-
space-time. There isn't a time slowdown as well because it can not change under
rotation of the coordinate system. There are only the light paths which length
changes as they are used as the coordinates in the coordinate systems. There is
the Lorentz contraction as a imaginary change of one coordinate at the expense
of another one in the course of rotation. By other words, the Lorentz
constraction is only projection.
We stress that the rotation used here is not inherent to the nature but is
involved to accommodate the point of view of unmoved observer with the moving
body. That is why the Lorentz transformation in contrast of the Galilean one
can't apply correctly to consideration of two reference frames moving each with
respect to other.
It is easy to get Galilean transformation from the picture proposed if we
release from the rotation. For this purpose one must associate moving point B
with the beginning of the second coordinate system as a moving reference frame.
Then the Galilean transformation may be put down as following
where x" is the distance to the point E in the moving frame and t" is the time
needed to the light to reach the beginning of second coordinate system from the
beginning of the first one. Again here v is the velocity of the second system
determined by the Eq.(1) and therefore its value can not exceed the velocity of
light. It is the only constraint put on the Galilean transformation in the case
of big velocities. Discrepancy with the Lorentz transformation seems may be
caused by the fact that it was used in fact only one reference frame experienced
the rotation around one point of view as it was shown above. If it is so, it is
not quit rightly to apply the Lorentz transformation to moving reference
frames.
2. The Michelson-Morly experiment
The Michelson-Morly experiment was a basic argument in favour of theory of
relativity. By simple interferometer which scheme is shown on Fig.2 it was
proven that there isn't the aether, a medium where proposed all matter might
move in. Namely, it was found that the paths of light going from mirror 0 to
mirror 1 and 2 are equal in any orientation and therefore are not dependant
from the movement of the Earth though the aether even if it exists. The result
of that experiment tried to explain either by Earth dragging of the aether or
by the Lorentz contraction along the direction of Earth movement.
To make the next step we must clarify what the aether is represented here
because now there are a lot of conceptions of this entity. Our vision on the
subject was presented
[1].
We imply that there are a 4D-medium consisted from
the tiny particles. The medium may be called the aether is closed and has a
bordor. There are vortices with different sizes in it represented the
fundamental particles. The light is a sort of perturbation on the 3D border
caused by the excited particles. An outing of the vortex in the form a mouth
determines the position of the particle. The body is the connected set of
vortices in this picture. That is why we (had been made from the vortices
themselves!) always see objects only in 3D space. The rotation previously
mentioned under the Lorentz transformation's derivation assumed to happen
on the border of the medium.
Hereafter we put for simplicity that the body is the 1D line, or the string,
and that the medium fills all the 4D-semi-space.
Now we try to show that if to think about aether as above the M-M experiment
may explain by the Galilean transformation. For that purpose we
consider the motion of the whole apparatus from the point of view of an
observer connected with the medium.
Let us suppose that the proposed motion proceeds along the x-axes. From the
rendering Fig.3, where the light path in transverse direction respecting this
motion, from mirror 0 to mirror 2, is shown, one may easily get the length of
this path
where l0 is the distance between mirrors 0 and
2 being measured at rest, i.e. when the interferometer is unmoved with respect
to observer. It is required a time interval to pass this distance
One should notice a resemblance between Eq.(9) and the first term of Eq.(3).
All distance passed by whole device for the time t2 is
To consider the light path in the longitudinal direction, from mirror 0 to
mirror 1, we cross the 4D-medium by the 2D-plane passing through these mirrors
represented by two strings as it shown on Fig.4. If the following assumption is
being made, it was quite easily to reach that purpose.
Namely, we suppose that the moving string is tilted with respect to the 3D
border of the medium. The tilt is not constant but exists only in the vicinity
of the border. So the string bents when it moves. The string situated normally
to the border rests because the forces of the surface tension at the outing
of the vortex are in equilibrium. Any slope of the vortex violates the
equilibrium and causes the vortex to move in the tilt direction. Moreover, the
value of velocity of the movement is assumed to be determined by the Eq.(1).
The more rigid grounding of this statement is to be published elsewhere. Here
we take it as an axiom. It is to note that the rotation used above under
the derivation of the Lorentz' transformation could be applied without
restrictions at any direction in the 3D-plane, or hypersurfice, situated
normally to axis x and came by the point A on the Fig.1. In other words,
we could use axis x4 instead y = x2. The rotations
in 4D-space are came indeed from the 2D-plane.
First of all we must know if the distance between tilted strings is changed or
not during their motion into the medium. Here one should distinguish a visual
distance in 3D space and a hidden one in 4D-space. Obviously, it should be more
strong and expansive force to make close two vortices in all 4D-space then
only on the border. So we postulate that the distance between two mirrors is
unchanged during their movement but the distance between their outings on the
border enlarges. This statement is in acute contradiction with the Lorentz'
contraction but lets easily to determine the distance between the mirrors 1
and 0 as l0 /cos α
It will be the same path for light to pass as in Eq.(9) if one should suppose
that in the moving reference frame the velocity of light is equal c, the
velocity of light in the rest frame. Such supposition is made in the special
theory of relativity, but we avoid it in favour of the Galilean transformation.
For this purpose We are to consider the details of the light path in the
reference frame connected with the medium.
Using the Galilean transformation the light path from mirror 0 to mirror 1 can
be represented as
|
l1' = vt1' + l0 /cos α
|
(12) |
and the reverse path from mirror 1 to mirror 0 as
|
l1" = -vt1" + l0 /cos α
|
(13) |
Their sum is the common path of the light and their difference is the common
displacement of the interferometer. The former gives the single solution
to reconcile with the result of the experinent:
The latter should be the same as s2 gotten from Eq.11.
It gives the correct meaning of the displacement only if the velocity
v satisfies Eq.1.
This result may mean that the velocity of light is not the same in the
different reference frames and differs from that in the reference frame
connected with the medium. As it is understood from the last equations the
light passes the distance l1' for the time instance t1'
with the velocity c+v and the distance l1" with the velocity c-v.
We may suppose the particles of light, photons, gain the additional impulse
while reflecting from the moving mirror 0 and loose it while reflecting from
mirror 1. On the other hand, the quantum mechanics demands the impulse of the
photon having been determined by its wave vector in accordance with the
Einstein's equation
without changing velocity of photon. Although there are no otherwise
evidences, it is left to surmise that there isn't any direct experiment
on measurement of the velocity of light glanced from a high-speed
moving mirror still.
One may try to explain the result by another manner. While the wave
length is declined after the first impact with the mirror 0 in accordance
with Eq.(16), its frequency is grown up to corresponding value so that
the velocity of light is not being changed. The inverse effect is
happened after the second impact at mirror 1, such that the common result
is vanished and therefore can't be detected by the interferometer. The
frequency alteration, however, might be fixed at the experiment
mensioned above.
It is interesting to compare this results with those getting from the assumption
of the theory of relativity. If the velocity of light is not changed after the
mirror thrusts the particles of light, we must put down l1' =
c t1' and l1"= c t1". Substituting these
values into Eqs.(13) and (14) we can easily determine the path gone off by
the light as
and the path gone off by the interferometer as
|
s1 = 2 l0 tan α / cos2 α
|
(18) |
The discrepancy with the experiment is obvious. Although the value 1/cos2 α on
which l1 differs from l2 is slightly differed from unity with small angles
α and small velocities v, respectively, it grows as 1 + (v/c)2 and
therefore might be determined by the experiment of Mickelson type. It seems
hardly might be explained the distinction in values s1 and s2 as well.
It would be note also that in view of sutuation stated above one need to
revise the concept of the reference frames in the case of 4D-space. The
moving frame should be tilted as well to accommodate the noving body.
3. The "wave-particle" problem
In [1] we are presented the model of the atom as a spiral line instead
of the point-like electron
wound along the line of nucleus. If such atom is in the excited state, some
loops of the spiral shrink and the excitation can move along the central line
of the atom in a similar way as solitonic excitation. When it reaches the border
of the medium the emission of the photon is happened. We may propose that the
form of the photon is a spiral line too. However, its location is not normal to
the border as for atoms but turned in such a way that its axis is directed
along the border. Such form is expected to promote the fast motion of the
spiral along the border of the medium. Therefore the photon can be considered as
a part of electron because it is produced from the electron. It can be
immersed by the atom with the specific sizes of the electronic system and
emitted with specific parameters as well. The more determined model is to be
published elsewhere. At that study one may detect that there isn't any
intrinsic distinction between the "wave" and the "particle". The both entities
are the same in scope of the 4D-model of the aether.
Moreover, if we consider the spiral as an atom of matter tilted with
respect to the normal to the border of the medium, i.e. the moving atom,
it is easy to notice that the loops of the spiral leave the waves on
the border. Their wave vector is dependent from the tilt angle α
as sin α or, as we know, velocity of the atom. It is in the
excellent accordance with the de Broyle formula for impulse of the
"wave-particle" in the form of Eq.(16). The coefficient of the
proportionality, the Plank constant ħ, is related to the geometric
parameter, the spiral spacing, which is determined by the properties
of the medium considered.
[1] V.Skorobogatov. The light in
4D-Aether model (in Russian). vps137.narod.ru/article2.html, 2005.
