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New ideas about the structure of reality, or
how to connect relative motion with an
absolute reference frame and describe
relativistic effects without Einstein’s
postulates
Witold Nawrot
February 12, 2024
https://doi.org/10.32388/3O4QMJ
Abstract
The paper presents a new approach to space-time problems that is completely
different from the approach used for over 100 years. The essence of the
changes are two new ideas that can be treated as a complement to the theory
of relativity. The first is the description of reality as a four-dimensional
Euclidean space. What we observe as space-time dimensions are directions in
objective (Euclidean) space, and these directions are not constant but depend
on the pair of bodies – the observer and the observed body. Depending on
the choice of body pair, the same direction in objective Euclidean space can
be interpreted as a temporal or spatial dimension of the observer’s coordinate
system. The new model allows the body to be described directly as a wave
and allows for a connection of the ideas of absolute space and the relativity
of motion. The second idea binds the transmission of signals (quanta) to
the systems of sending and receiving particles. As a result, the motion of
the quantum is always constant in the system of the sending and receiving
particle. This justifies the constancy of the speed of light regardless of the
relative velocity of the bodies. Quanta are no longer independent particles
but are disturbances of particles, which are treated as waves.
The proposed changes simplify the description of relativistic phenomena,
eliminate the need to apply Einstein’s postulates by introducing mechanisms
describing the relative motion and propagation of quantum, bind the descrip-
tion of relativistic and quantum phenomena by describing bodies directly as
waves, extend the range of phenomena described within one model and solve
many problems impossible to solve within the theory of relativity.
The paper compares the descriptions of particular problems in the Theory
of Relativity with the descriptions of the same issues in the new model. In
most cases, the predictions of both models are similar, but the differences
in the construction of the models give different conclusions in some cases,
which is the basis for proposing specific experiments allowing verification of
the proposed approach. Some of the proposed experiments can be carried
out with the use of existing experimental devices.
1Introduction
One of the fundamental problems in the development of physical models,
which has already been considered in the past, is the question, valid prac-
tically at every stage of the development of sciences, about the relationship
between the coordinate system that we use to describe reality and the actual
shape of this reality.
When creating a new model of reality, we initially use the coordinate
system that we use every day to describe our immediate surroundings to
describe reality. However, it may happen that the coordinate systems we
have used thus far are not an appropriate tool for describing all phenom-
ena and correctly describe only a certain class of phenomena occurring in
our immediate environment. The use of an improper coordinate system may
lead to obtaining correct results, but at the cost of excessive complications of
the mathematical description and the need to introduce all sorts of nonintu-
itive assumptions, the only justification of which is to obtain correct results.
This has already happened in the past, for example, in the case of the geo-
centric model, when a coordinate system was used to describe the motion
of the planets, previously used only to describe phenomena on the Earth’s
surface. The use of an improper coordinate system – bound to the Earth
and not to the Sun – allowed us to obtain correct results consistent with
astronomical observations, but at the cost of the abovementioned necessity
to significantly complicate the mathematical description and introduce such
creations as Epicycles, Deferents and Equants – artificial, nonintuitive and
forgotten concepts, but successfully used to predict the position of the plan-
ets in the firmament for over 15 Centuries. The obtained model was overly
complicated and referred to unnatural body motions that we had not ob-
served in our surroundings and that did not conform to the laws of body
motion known at the time.
Such historical experience regarding errors in model construction should
sensitize us to cases when the model we are currently constructing begins
to be overly complicated. In such cases, it would be advisable to check
whether the complication of the model does not have its source in erroneous
assumptions made during the construction of the model, even if the model
in its current complex form allows us to obtain correct experimental results
confirming the predictions of this model.
Between 1905 and 1916, the theory of relativity, describing relativistic
phenomena, was created. The model was based on dimensions that deter-
mine the temporal and spatial distances that we measure in our direct sur-
roundings. The addition of the fourth dimension to the existing model of
three - dimensional space made it possible to treat four - dimensional space-
1
time as an integral whole, however, at the cost of a significant complexity
of mathematical description. While three-dimensional space is a Euclidean
space, four-dimensional uncurved space-time is a pseudo-Euclidean space,
in which distances are calculated according to different rules than in three-
dimensional Euclidean space. This greatly complicated the mathematical
apparatus necessary to describe reality.
The theory of relativity has been working correctly for over 100 years and
has been experimentally confirmed many times, but this does not remove all
doubts about the construction of the model. The fact that the construction
of Minkowski’s four-dimensional space-time is much more complicated than
the construction of the three-dimensional Euclidean space in which we live is
at odds with the conviction of a part of the scientific community, confirmed
many times during the development of science, that reality takes the simplest
possible forms. Therefore, if reality is four-dimensional, then perhaps it
should also be Euclidean.
This belief has inspired a relatively large number of papers aimed at
describing space-time with a four-dimensional Euclidean model of spacetime
and pointing out errors and inaccuracies in the Theory of Relativity that
have not been taken very seriously by reviewers [1-56,58], but nevertheless,
some of these papers have been published in reputable journals. However,
unsuccessful attempts to describe reality with the help of the Euclidean model
of four-dimensional space-time have led to the conviction that the Euclidean
model of reality built from the dimensions of time and space is impossible to
construct.
However, there is still an open question about the relationship between
the space-time dimensions used to describe reality and the actual dimensions
that make up reality. We are now describing reality with the Minkowski
space-time model, which excludes the existence of a description independent
of the observer’s system. Nevertheless, I wanted to try to analyze what
transformations need to be made in the Euclidean space, which we are used
to when observing our immediate surroundings, to obtain a description of
spacetime that satisfies the equation for the conservation of the space-time
interval.
2
2Thetworealitymodelsconservingthespace-
time interval.
2.1 Objective and observed reality
Unlike the creators of the Theory of Relativity, who, when creating a new
model, were only determining its features, which then needed many years for
experimental confirmation, we are richer in knowledge about the properties
of space-time, which must be preserved independently of the model of reality
underlying the space-time description, and thanks to this, we can look at the
problem of the construction of space-time from a certain distance, which was
not available to the creators of RT.
First, we will separate the concepts – objective coordinates – describ-
ing hypothetical reality as it actually is, regardless of what we observe and
measure – let us call it "objective reality" and the reality we observe, built
from the temporal and spatial distances that we record with our measuring
devices – let us call it "observed reality".Ofcourse,accordingtothetheory
of relativity, the objective reality that allows the description of phenomena
regardless of the choice of the observer cannot be determined, but this issue
will be discussed later in the article.
We will assume that the "objective reality" is described by the four-
dimensional Euclidean space E4 with dimensions a1,a2,a3,a4;howeversince
most of the discussed issues can be reduced to two dimensional problems in
practice, for the time being, to simplify considerations, it is enough to zero
two dimensions in E4 and analyze only its two-dimensional subspace E2 built
from points with coordinates a1
i,anda2
ithen, if necessary, we will expand
the solutions for E4 space.
In E2, we can specify certain distances, but at first, we should not assign
the meaning of temporal or spatial distances to these distances. These are
simply distances, and whether we express them in meters or seconds – this
will result from future considerations. In practice, the representation of such
a space can be the plane of our drawings. In the Euclidean subspace E2, we
can determine the distances between its points, but these distances do not
depend on the choice of the coordinate system, which can be constructed
from axes determined by any two linearly independent vectors (Fig. 1). In
this case, if we have the axes of the coordinate system in E2 marked as a1
and a2inclined to each other at an angle ↵,thenthedistancebetweenthe
two points s in E2 is equal to:
s2=(a1)2+(a2)22a1a2cos(↵)(1)
3
Figure 1: Euclidean space can be described by any coordinate system deter-
mined by linearly independent vectors.
Since the choice of angle ↵does not affect the value of the distance s,
to simplify the calculations, it is most convenient to use the rectangular
coordinate system -see Fig. 2. Taking the angle ↵=90
0,thedistancecan
be calculated in the simplest way:
s2=(a1)2+(a2)2(2)
Figure 2: In Euclidean space, the choice of orthogonal coordinates is dic-
tated solely by the simplification of distances describing formulas and the
associated convenience of use.
4
At the same time, it is not a form resulting from the properties of the E2
space but only a form that is more convenient for calculations and directly
corresponds with the environment that we observe every day.
Therefore, to begin with, we have an "objective reality" that we can de-
scribe as Euclidean space and an orthogonal coordinate system in this space
defined by axes, which we will denote further as a1and a2.Livinginsuchan
"objective reality", i.e., for now, in the space of E2, we measure the tempo-
ral and spatial distances between events. The measured distances and times
create the picture of reality that we observe. We will call it "observed real-
ity".Distancesaredefinedinourreferenceframeusingt,x coordinates. For
simplicity, we write the dimensions of time and space in the form ct,x, which
means that the dimensions of both time and space are presented in the same
units. In this case, we can define these dimensions as x0(= ct)and x1(= x),
but for now it will be more illustrative to use the existing designations: ct, x.
Now, we will try to determine the relationship between the "observed
reality" described by the coordinates ct, x and the objective reality described
by a1,a
2.
If we now want to describe the nonrelativistic relations of the position in
space to time, then as in the case of the description of distances in the space
E2, we can choose the axes ct and in E2 inclined to each other at any angle,
but then again we have to use the inconvenient conversion of the measured
values of the coordinates of time and space that make up the "observed real-
ity" into the coordinates of the space E2 that define the "objective reality".
In addition, we have related difficulties in determining the relationships be-
tween the dimensions of space E2 and the observed dimensions of time and
space because at any orientation of the ct-andx-axis, the dimensions a1
and a2are different combinations of dimensions of time and space, and the
description of objective reality is very different from the observed reality –
Fig. 3.
5
Figure 3: In the most general case, the dimensions of time and space do not
have to be perpendicular to each other.
However, it does not have to be that way. It is enough to select the x-axis
along the a2axis of objective reality, for example, and the ct time axis along
the a1axis. Thus, now, the coordinates describing objective reality a1and
a2correspond to the observed coordinates ct and x,respectively–Fig.4.
6
Figure 4: In a nonrelativistic reality, we assume that the dimensions de-
scribing objective space are identical to the dimensions of time and space
describing the observed reality.
With such a choice of coordinates, the "observed reality" is identical to
the "objective reality",butthischoicewasdictatedonlybythegreatestsim-
plicity of the description of reality, which does not exclude other orientations
of the ct-andx-axis in relation to E2 - Fig. 3 - omitted only due to the
greater complexity of the model construction. Thus, we now use a system of
"observed coordinates" to describe reality, and we can assume that "objective
reality" –independentoftheobserver–existsandisrealityexactlyaswe
see it. However, this form of reality correctly describes only nonrelativistic
cases.
In summary, for a nonrelativistic model, we can assume that there is an
"objective reality" independent of any observer, whose dimensions a1and
a2are orthogonal and identical to the dimensions ct and xregistered in the
observer’s coordinate system.“Objective reality” ,independentoftheobserver,
means that, for example, we can describe such an “objective reality” using the
concept of the Aether – the medium that fills such reality and allows us to
determine an absolute reference frame.
7
2.2 The first reality model conserving the value of the
space-time interval
If we want to include relativistic cases in the description, we must consider
the fact that space-time distances must satisfy the principle of conservation
of the space-time interval. Therefore, we need to modify our description so
that the time and space coordinates for the observer and the observed body
satisfy the equation for the conservation of the space-time interval:
c2dt12dx12=c2dt22dx22(3)
We will now base our reasoning on one of the applications of the space-
time interval – when we limit the space-time interval to the description of
the mutual observation of two bodies. This limitation will allow us to draw
some conclusions that will allow us to take a new look at the problems of
space-time interval and coordinate transformation.
Let us consider two bodies with coordinate systems ct1,x
1and ct2,x
2
moving relative to each other in space-time. If the observed body is the
body ct2,x
2and the observer is the body ct1,x
1,andsinceintheobserved
system dx2=0, our equation will take the form:
c2dt12dx12=c2dt22(4)
On the other hand, if we take as an observer body 2 - then dx1=0-and
our equation looks like this:
c2dt12=c2dt22dx22(5)
If we start from the model of nonrelativistic reality described above, in
which the time coordinate is perpendicular to the spatial coordinate, then
for these equations to be satisfied, the time axis of the coordinate system
of the body assumed to be observed must be stretched, and a slightly more
detailed analysis shows that the space axis of the system of the observed
body should also be deformed. Deformations of the time axis depending on
the observer’s choice for exemplary values of coordinates equal to 3, 4, and
5 are shown in Fig. 5, while a general scheme describing the deformations of
space-time dimensions in the system of the body observed as a function of
velocity relative to the observer’s system is presented in Fig. 6.
8
a b
Figure 5: Deformations of the coordinate system time axis depending on the
observer’s choice. In Fig. a the observer is the body ct1,x
1, while in Fig. b
the observer is the body ct2,x
2. Deformations are shown for exemplary values
of temporal and space coordinates equal to 3, 4, and 5.
Figure 6: Deformations of coordinate systems of the observed body ct0,x
0as
afunctionofvelocityrelativetotheobserverct, x.
However, the problem of deformation of the coordinates of a body in
motion is symmetrical for both observers, and the swapping of the observer
9
causes deformations of the coordinates of the body previously considered
an observer. Thus, if the deformation of the coordinates ct, x depends on
the choice of the observer, then the problem arises whether it is possible to
determine a coordinate system a1,a
2common to all observers and at the same
time allowing us to determine the coordinates ct, x dependent on the choice
of the observer. Lorentz tried to tackle this problem by creating the theory of
the Aether, but it generated many other problems. The problem was solved
in 1905 by Einstein by simply rejecting the existence of such a structure as
objective space independent of the observer – in this case, the E2 space – and
limiting the description only to space-time coordinates ct, x, which describe
the time and space distances measured in the system of a particular observer.
Consequently, Einstein accepted all motions as relative. Of course, in his first
works, Einstein did not yet know the principle of conservation of space-time
interval, which we apply here already at the basic stage of considerations, but
as I wrote at the beginning, we allow ourselves to use information unknown
to the authors at the time of creating the Theory of Relativity.
In this way, Einstein solved the problem of the existence of the Aether,
which simply ceased to be necessary to describe reality.
To sum up:
According to the first approach described above, the space-time dimen-
sions that determine the time and space distances we observe are treated as
dimensions that create reality. The space-time dimensions of the observer co-
ordinate system are orthogonal coordinate systems and must be associated
with the featured observer. A description of reality is possible only from
the system of the distinguished observer. There is no objective reality or
coordinate system in which phenomena can be described regardless of the
observer’s choice. The principle of conservation of the space-time interval is
the result of the deformation of the axes of coordinate systems of bodies in
motion.
Let us now try to approach the issue from a completely different point of
view, which was not available before the publication of Minkowski’s works. In
fact, the second approach presented below could not have been published and
developed until after the de Broglie hypothesis was announced in the mid-
1920s, but that was the time when the Theory of Relativity was triumphant,
and any revision of the theory was probably of no interest to anyone at the
time.
10
2.3 The second model of reality that conserves the value
of the space-time interval
As I wrote above, if we assume the existence of ”objective reality” –here
described for a moment by means of the space E2 with coordinates a1,a
2,
then theoretically, as I mentioned before, we can assume that the ct, x axes
can be selected not necessarily as perpendicular to each other (Fig. 3).
The choice of perpendicular axes is consistent with the image of reality that
we observe locally in our immediate nonrelativistic environment, in which
right angles are dominant in architecture and result from the direction of
gravity perpendicular to the plane of the Earth’s surface. If we observe
width, height, and depth and describe them mathematically as perpendicular
dimensions, which is the most convenient way of describing them from our
point of view, then when we add a fourth dimension to the description,
we also take for granted that it should be perpendicular to the other three
dimensions. However, as I mentioned earlier, the space-time axes ct, x can
be represented on the E2 plane as axes inclined to each other at arbitrary
angles (Fig. 3).
Therefore, if we allow ourselves to freely draw the axes ct, x on the plane
described by the (objective) space E2, we will notice that the fulfillment of
the space-time interval equation in the form described by equations (4,5) for
the case of mutual observation of two bodies will be possible, also when the
space axis of the observer’s coordinate system is chosen as perpendicular to
the time axis of the coordinate system of the observed body.
In that case, the time axis of the reference frame of the observed body
and the time and space axes of the observer form a right triangle in E2 -
(Fig. 7).
11
a b
Figure 7: Axes of space-time reference frames of the observer and the ob-
served body presented in the E2 space. In Fig. 7a, the observer’s system is
the axes of body 1 - ct1,x
1, while in Fig. 7b, the observer’s system is the
axes of body 2 - ct2,x
2.
From this r i g ht triangl e , t h ey immediately d e r ive:
1. Definition of relative velocity in the form:
V=x1
ct1
=x2
ct2
=sin'(6)
and
2. dependencies (4) – Fig. 7a and (5) – Fig. 7b
which can now be written in the form, respectively:
ct2=ct1cos'=ct1p1sin2'=ct1p1V2(7)
and
ct1=ct2cos'=ct2p1sin2'=ct2p1V2(8)
In addition, I would like to emphasize that while in Minkowski space-
time we could present the observation only in the reference frame of one of
the bodies, in E2 we can assemble both observation schemes from Fig. 7a
and 7b and present them in one drawing in the space E2, where we can see
12
simultaneously the diagram of observation of body 1 by body 2 and vice
versa – body 2 by body 1 – Fig. 8. Thus, the change in the reference frame
conserves the coordinates of the bodies in E2, i.e., the space E2 (and of course
E4) can be treated as an objective and at the same time absolute reality –
nonexistent in the previously presented approach described by Minkowski
spacetime.
Figure 8: In E4 we can represent the mutual observation of bodies in a way
that is independent of the observer’s choice. There are 2 orthogonal directions
associated with each body (the four orthogonal directions in E4) – the first
direction interpreted by this body as the time axis and the second direction
perpendicular to the time axis of this body, which in turn is interpreted as the
space-dimension (three space-dimensions in E4) of the coordinate systems of
all observers observing this body. A change in the relative velocity of the
observers means the rotation of one of the orthogonal systems relative to the
another by an angle '
Note that regardless of the relative velocity of the observers – that is,
the sine of the angle between the time coordinates of the two bodies, in
the case of mutual observation of two bodies – we always have two systems
of rectangular coordinates: the first ct1x2composed of the time coordinate
13
of one of the bodies and the direction perpendicular to this time direction,
which the second observer interprets as his space-axis x2,andthesecond,
an analogous ct2x1composed of the other two orthogonal dimensions. The
change in the relative velocity of these systems consists only of the rotation
of one such orthogonal system ct1x2relative to another ct2x1by an angle
that is a measure of the relative velocity of one such system relative to the
other (6).
Such a concept of Euclidean space – the so-called "mixed spaces" –was
proposed in an article by A. Gersten [1] in 2003. However, we can see that
the dimensions defined in this way cannot be the dimensions that create the
"true" reality – as A. Gersten assumed in his article. For the choice of space-
time axes shown in Figs. 7 and 8 to make sense, we must assume that the
space-time axes of both bodies depicted in the drawings in objective space E2
are not dimensions that create reality but only directions in E2 (and in fact
in E4), which we interpret as space-dimensions and a time-dimension when
observing the bodies around us. However, these directions are not constant
but depend on both the observer and the body observed.
As mentioned above, it is clear from Fig. 8 that there are two orthogonal
directions associated with each individual body in E2: the direction that de-
termines the time-axis of the observed body and the direction perpendicular
to the time-axis of that body, and this direction is interpreted as the space
direction when observing that body by all observers observing that partic-
ular body. The case presented in Figs. 7 and 8 is already a representation
of the space-time axes as directions in objective space E2. In E4, the space
direction shown in Figs. 7 and 8 actually corresponds to any direction in
subspace E3 perpendicular to the time axis of the body observed in E4.
The topic of observation will be developed in the discussion of the concept
of space and on some other topics.
Therefore, let us summarize the assumptions of the second model.
According to the second model, E4, there is an “objective reality”, which
is a four-dimensional Euclidean space in which distances do not have a pre-
determined meaning of time- or space distances. Time and space distances
are measured along certain directions in E4, which we interpret as tempo-
ral or spatial dimensions that are not constant but depend on the currently
observed body. The directions interpreted as the space dimensions of the
observer are perpendicular to the time axis of the coordinate system of the
currently observed body, i.e., they are different for the observation of different
bodies. Despite the existence of "objective space",i.e.,inpractice,anabso-
lute coordinate system and the related absolute nature of the coordinates of
bodies in E2, the determination of relative velocity, which is a measure of the
angle between the directions of the observers’ time axes, allows motion to be
14
defined as a relative value measured solely depending on the selected body.
The mechanism responsible for the conservation of the space-time interval is
as follows: in practice for the occurrence of relativistic effects, is the change
of directions in E4 (in Figs. 7 and 8 in this work – in E2), interpreted as
space-dimensions for the observation of various bodies. Note that the in-
troduction of an absolute coordinate system in space E2 (in practice in E4)
will also allow us to return to the Aether hypothesis, which will be discussed
later.
Thus, in practice, we have two alternative models – the Minkowski space-
time model and the four-dimensional Euclidean space model – let us call
it the "E4 model" – both of which justify the principle of conservation of
the space-time interval, but for the time being only to the extent limited
to the mutual observation of two bodies. Since the concept of space-time
interval is broader than the dependencies underlying the above reasoning
(4,5), in the following chapters, we will compare the interpretation and scope
of application of concepts such as time, space, motion, and the scope of
application of the space-time interval in both models presented.
If both models satisfy the same relationships resulting from formulas (4,5),
then the same solutions to problems related to relativistic problems can be
expected in both models – Minkowski and E4. However, the different concep-
tions of reality underlying these models, the much wider scope of application
of the space-time interval equation in the theory of relativity than that re-
sulting from formulas (4,5), and the different understandings of some basic
concepts lead in some places to different results, which in turn can be used
for experiments confirming or negating any of these models.
3InterpretationofdifferentconceptsintheMinkowski
model and in the E4 model
3.1 Time
3.1.1 Minkowski space-time model
Time in Minkowski’s space-time is the fourth dimension of space-time, but
it is a difficult concept to interpret. The flow of time can be imagined as the
motion of a body along the time dimension in four-dimensional space-time.
However, this "motion" depends on relative motion and on the gravitational
field, which change the scale of the time dimension (Fig. 6), making it
virtually impossible to unambiguously interpret the flow of time as the motion
of bodies in four-dimensional reality. The lack of a precise understanding of
15
the concept of time leads to various speculative conclusions, such as going
back in time, which could theoretically occur if time as a dimension were
consistently treated. Aristotle, on the other hand, treated time as motion.
However, for the time being, the theory of relativity, based on the Minkowski
spacetime model, does not definitively explain the problem of time. Instead,
it gives us the tools to accurately determine the impact of the relative motion
of bodies and the curvature of space-time on the running of clocks (which is
used on a daily basis, for example, in GPS systems), and it makes a significant
contribution to technological progress.
3.1.2 Model E4
An alternative description of reality proposed by the E4 model allows for a
more precise determination of the nature of time.
If we assume that the flow of time can be represented as the motion of
a body along its time-axis in E2, then because, according to Fig. 7, 8, the
choice of the observer does not affect the change of the properties of the
time-axis of the body, such as the direction or scale of the axis in E2 and
consequently in E4 (regardless of whether we are talking about the time-axis
of the observed body or the observer), so the motion of the body along its
time-axis can be interpreted directly as absolute motion in E4. Since there is
no reason to treat different bodies differently, we can assume that the motion
of all bodies along their trajectory in E4 is constant, and the easiest way is
to assume that this motion takes place with the velocity V=1.Inthiscase,
the proper time of a given body can be identified with the path that this
body travels in E4.
Since all dimensions in E4 are on the same scale and the distance traveled
by the body in E4 is equal to the time indicated by the body clock, it is no
longer advisable to mark the time-axis by ct.Fromthispointon,wedenote
time only as t, and of course, this will further lead to the fact that the speed
of light is also equal to unity here – as will be discussed in the following
chapters. Thus, in the following part of the text, the notation in which
c=1will be used, both in relation to the E4 model and in relation to the
Minkowski spacetime model.
We do not consider here non inertial motions, for which the distance
traveled in E4 and the elapsed time in the system of the particle will differ,
but in this article, we deal only with inertial motions.
Unlike Minkowski’s model, the time-axis of a body is not a dimension
but merely the path along which the body moves in E4 (similar to a train
moving on tracks where the tracks play the role of a time-axis). Therefore, for
example, going back to your previous position in E4 does not mean traveling
16
backward in time, because the time is the way you have traveled in E4
regardless of the direction of motion. The body clock in E4 plays a role
similar to the odometer in a car, which does not return when we turn around
and drive in the opposite direction to the previous one. Consequently, in E4,
the notion of time travel loses its meaning.
It is a pity . . .
Thus, the particle can be imagined as moving – in this article only along
rectilinear paths (trajectories) in E4 with a constant velocity V=1. This
is a motion relative to E4, i.e., an absolute motion. With the motions of
the bodies in E4 along rectilinear trajectories, the distance of any two points
on the trajectory in E4 is the difference in the times indicated by the body
clock at these two points in the trajectory. Thus, for the time being, we can
deal with the definition of time as the distance between points in E4 without
introducing the additional concept of absolute time, in relation to which we
would determine the velocity of the E4 bodies – which will prove necessary
in the case of non inertial motions – not discussed in this article.
The problem of mutual observation of time dilation does not lie in the
deformation of the time dimension but in the change of direction interpreted
as the space-dimension of the observer, which causes the observed path (time)
of the body in motion in E4 to seem shorter than in the observer’s system
– this is shown in Fig. 7 and formulas (7,8). As long as the bodies are
moving in rectilinear paths, this effect is symmetrical, and both observers
observe an identical dilation of time in the system of the other body. For
the mutually symmetric time dilation in a system of bodies to become an
actual contraction of time, one of the bodies must change the velocity (value
or direction), as described in [54,55].
In summary, the bodies in E4 move along certain paths at a constant
velocity relative to E4. The direction of motion of the bodies in E4 is inter-
preted by the bodies as the time-axis of their coordinate system, while the
very fact of motion in E4 is interpreted by the bodies as the flow of time.
In E4, it is not possible to distinguish a single time dimension common
to all bodies. Therefore, there is no such thing as a dimension of time. The
time dimension in the E4 model is only an observed value.
3.2 Space
3.2.1 Minkowski space-time model
If we have a particle producing a field, then the field permeates three-
dimensional space, and the field causes interactions to occur. Thus, we can
define a space in Minkowski spacetime as a subspace in which interactions
17
propagate. Since we know reality by means of interactions, we observe only a
three-dimensional subspace when we interact with bodies. If the interactions
propagate in the fourth dimension – time – we could interact with events in
the future or the past, i.e., simply observe both past and future events, but
we do not observe such interactions.
Thus, space made up of space distances that we measure with the methods
available to us is an integral component of space-time. Space dimensions are
identical to the directions of propagation of interactions in Minkowski space-
time. The observer uses the same spatial coordinate system to observe all
surrounding bodies.
3.2.2 Model E4
As stated above, according to the E4 model, the directions interpreted by
the observer as space dimensions are perpendicular to the time axis of the
body (trajectory) of the observed body. Since we identify space dimensions
with the directions of propagation of interactions, the above sentence means
that the interactions are emitted perpendicular to the trajectory
of the observed body. The fact that, as I wrote previously, there are
two groups of directions associated with each body: the direction of motion
interpreted as the time-axis of the body and 3 directions perpendicular to the
direction of motion of the body interpreted as space-directions common to all
observers of this body, allows us to represent the body as moving in E4 with
an absolute velocity equal to unity and emitting interactions in directions
perpendicular to its direction of motion in E4.
18
Figure 9: Three bodies watching each other. Each body interprets a different
direction in E2 as its space dimension when observing other body. If one
observer, e.g., tA, observes two bodies tB and tC, then observing each of
these bodies interprets a different direction in E2 as the space-dimension of
its coordinate system – these are directions perpendicular to the time-axes of
the observed bodies – x2and x3,respectively. Inturn,allobserversobserving
the same body interpret the same direction (perpendicular to the trajectory
of the observed body) as their space dimension. For example, both observers
tAand tBobserving the body tCtreat the x3direction as the space-dimension
of their coordinate systems.
As seen in Fig. 9, when one observer observes several particles, he in-
terprets the different directions in E4 (in Fig. E2) as his space dimensions.
The spatial direction shown in Fig. 9 as a one-dimensional line represents, in
E4, the subspace E3 orthogonal to the time axis of the observed body. The
only information that reaches the observer is information about the motions
of the bodies, which the observer observes along directions perpendicular to
the time axes of specific bodies. The observer builds a picture of reality by
combining observations of different bodies made at different points at dif-
ferent times into one common image. It can be compared to observing an
image on a computer monitor, where observing millions of pixels shining in
different places and at different times, we see one coherent and stable image.
Due to the absence of any absolute observable frames of reference, the
observer is unable to notice the differences between these directions in E4,
and these differences are perceived by the observer as relativistic effects.
However, living in a nonrelativistic reality on a daily basis, we observe
surrounding bodies as moving along parallel trajectories in E4, and then the
directions perpendicular to the trajectories of these bodies, interpreted as
space-dimensions, overlap with each other. Thus, in practice, we observe
19
reality as Euclidean four-dimensional space-time, and this image of space-
time determines our idea of the structure of reality.
In summary, there are no space-time dimensions in E4. The con-
cepts of time and space are the result of observation, not the actual construc-
tion of reality. Which of the E4 directions are interpreted by the observer
as three-dimensional space is determined by the choice of the observed body.
The observer selects a different set of directions in E4, interpreted as the
space-dimensions of his coordinate system, for each observed body.
We learn ab ou t t h e existence of spac e by analyzing t h e m o tions of the
bodies around us. These bodies emit interactions in directions perpendicular
to the directions of their motion in E4 (i.e., to the time axis of the system
of the observed body), and in turn, we are able to observe the motions of
these bodies along the directions of propagation of the interactions. Hence,
when observing different bodies, we interpret the different directions in E4
as space dimensions.
The discussed mechanism referred to the most common way of conducting
observations – by means of the exchange of interactions carried by energy
quanta, which makes us able to observe the reality around us. However, as I
have shown in my other works [56], for very small distances, it may be difficult
or even impossible to determine the spatial direction, and in such cases, the
whole process should be considered only in E4 coordinates, and only the
result obtained in this way can be described using space-time coordinates.
3.3 Motion
3.3.1 Minkowski space-time model
In Minkowski’s space-time, there is one concept of motion – it is a change
in the position in space of a body or quantum as a function of time. The
definition of velocity is the same for light quanta as for nonzero mass bod-
ies. This made it necessary to introduce a mechanism aimed at reconciling
the variable speed of mass bodies with the constant speed of light quanta.
Initially, it was the theory of the Aether, but Einstein presented a model
without the use of the Aether, which was made possible by recognizing the
relativity of motion.
In summary, motion in Minkowski space-time is defined as a change in
position in space with respect to time. Motion is specified identically for all
objects, regardless of their type. This motion is relative, i.e., it can only be
determined with respect to another body.
20
3.3.2 Model E4
The E4 model assumes the existence of an absolute Euclidean coordinate
system defined by the E4 space.
In the E4 model, there are three types of motions:
1. The absolute motion of bodies along their trajectory in E4. In the
case of rectilinear trajectories, it is an absolute motion with a constant
velocity V=1. This motion is perceived by observers as the flow
of time, while the distance traveled in E4 along the trajectory (for
rectilinear trajectories) is a measure of the time that has elapsed in the
coordinate system of the body.
2. Relative motion of bodies with nonzero mass. What we observe as
the relative velocity of the bodies is the sine of the angle between the
trajectories of the two bodies – this is shown in Fig. 7. The velocity is
described by formula (6). Since there is no distinguished direction in
the absolute space E4, the angle between the trajectories, or in other
words, the velocity of the body, can only be determined in relation to
the trajectory of another particle, i.e., the angle defined in this way,
and thus the velocity, are relative, regardless of the fact that the space
E4, in which we define these angles, is absolute.
3. Motion (propagation) of interactions. If bodies send interactions in a
direction perpendicular to their trajectories, and we assume (which I
will justify later) that in the rest frame of the body, the velocity of
propagation of these interactions is equal to Vint =1(in a system in
which c=1), it means that the resultant motion of the interactions in
E4 is the result of the combination of two perpendicular components
of the velocity: the aforementioned velocity Vint =1and the velocity
of the body along its trajectory in E4 equal to V=1.Fromthecom-
bination of these two velocities, it follows that the trajectory of the
light quanta (not the world line, because in E4 there is no concept of
space-time as a property of reality) is always inclined at an angle of
450 to the trajectory of the body sending the interaction. This means
that the speed of propagation of the interactions is always constant in
the coordinate system of the particle sending or receiving the interac-
tion. Fig. 10 shows a diagram of measuring the speed of light in a
laboratory setup. For parallel trajectories, the change in the speed of
the measuring system, which is the rotation of this system in E4, does
not affect the measurement result or the propagation of interactions
between the elements of the measuring system. Thus, an attempt to
21
determine the influence of the velocity of the measuring system, defined
as the angle between trajectories, on the velocity of quanta (e.g., in the
Michelson–Morley experiment) cannot give any results. The relative
speed and the speed of light are the result of separate phenomena, de-
scribed by different mechanisms, which is also shown in Fig. 10. Fig.
10 shows the case of measuring the speed of light for parallel trajecto-
ries, i.e., the case in which all elements of the measuring system remain
at rest relative to each other. The case for nonparallel trajectories will
be discussed later in the article.
a b
Figure 10: Fig. 10 Measurement of the speed of light in a laboratory system.
The light source and detector are in one place. A pulse of light emitted from
the source is reflected offthe mirror and goes to the detector. In Fig. a the
measuring system is shown "at rest", while in Fig. b the measuring system
moves relative to the system in Fig. a at a speed of 0.966 c. Note that while
in the case of the trajectory of light, the velocity V=1corresponds to the
trajectory of light inclined to the trajectory of the laboratory system at an
angle of 450, in the case of relative motions of bodies with a nonzero mass,
the velocity V=1corresponds to the motion along a trajectory inclined at
an angle of 900. This illustrates the difference between the types of relative
motion of non-zero-mass bodies and the motion of quanta.
To sum up: In E4, we have three types of motions – absolute motion
relative to E4 perceived by the body as the flow of time, relative motion
defined by the angle between the trajectories of bodies moving relative to
22
each other, and the third – the motion of signals/quanta by means of which
we observe reality, which is absolute motion along trajectory permanently
related to the trajectory of the body sending/receiving the signal (more on
this later in the article).
3.4 Space-time interval
3.4.1 Minkowski space-time model
In analyzing the fulfilment of the equation for the conservation of the space-
time interval, we have limited ourselves to the form of the interval describing
the mutual observation of two bodies (4,5).
The general equation for the conservation of the space-time interval for
the plane case for multiple ti,x
iobservers (in the coordinate system where
c=1)hastheform:
dt12dx12=dt22dx22=... =dtn2dxn2(9)
However, as you can see, the concept of interval in Minkowski spacetime
has a broader meaning and can generally refer to any distance ds in space-
time. The equation for the behavior of the space-time interval written in
the form (9) can be applied to any coordinate systems of observers in space-
time moving relative to each other. The object of observation of all these
observers can be any hypothetical space-time distance ds not necessarily
related to the observation of a particular body. In this case, the space-
time interval equation simply describes the deformation of space-time as a
function of the motion of any observer of the event described by the ds value.
The deformation of dimensions ensuring the conservation of the space-time
interval as a function of velocity in Minkowski spacetime is shown in Fig. 6.
What is important is that the Minkowski spacetime model does not provide
any additional constraints or conditions for equation (9).
In summary, in Minkowski space-time, the principle of conservation of
space-time interval applies to the conservation of any distance in spacetime
when the frame of reference changes.
3.4.2 Model E4
In the case of the E4 model, space or time dimensions are not integral com-
ponents of physical reality. Space-dimensions are only observed directions in
E4, perpendicular to the trajectory of the observed body.
To be able to recor d t i me dimensions, we need to know t h e t ra je c t ories
of the observers in E4. Determining the space dimension, on the other hand,
23
requires the choice of the observed body because without knowing the tra-
jectory of the observed body, we cannot determine the space dimensions of
the observers (which must be perpendicular to the trajectory of the observed
body). Thus, in formula (9), ds must be equal to the proper time of the
observed body dt0,andthespaceaxesofallobserversofthesamebodyt0
must overlap.
In the Minkowski model of space-time, we had space dimensions deter-
mined by the observer’s coordinate system and independent of the fact of
observing a particular body. Thus, the space-time interval equation could
describe any spacetime distance ds not necessarily related to the selected
observed body.
Meanwhile, the condition linking the direction, interpreted as spatial, as
perpendicular to the trajectory of a particular observed body causes equa-
tion (9) to take the form:
dt12dx12=dt22dx22=... =dtn2dxn2=dt02(10)
In other words, equation (9) has the additional limitation dx0=0.
This condition restricts the use of the space-time interval to observa-
tions of the distinguished body only. Since it is not possible to determine
space-dimensions in E4 without indicating the observed body, the space-time
interval equation can now not determine any arbitrary space-time distance
but only the distances lying on the trajectory of the selected body.
A graphical representation of Equation (10) is shown in Fig. 11.
Figure 11: The equation for the conservation of the space-time interval in E4
concerns the observation of the selected body by different observers. Accord-
ing to the fact that as a space-dimension we interpret the direction perpen-
dicular to the trajectory of the observed body and the directions interpreted
by all observers as their space-dimensions (at the time of observing the body
t0), coincide.
24
To sum up: In E4, since the selection of the observed body is needed
to determine the directions interpreted as space-dimensions, the space-time
interval equation makes sense only for the observation of that body. In the
empty space E4, where space-dimensions cannot be determined, the space-
time interval equation makes no sense.
Apparently, the difference between equations (9) and (10) is small, but
its consequences will be presented in the next chapter.
3.5 Coordinate transformation.
3.5.1 Minkowski space-time model
In Minkowski space-time, a space-time interval can refer to any distance in
space-time. The LT is a solution to the equation for the conservation of
space-time intervals for SR and has the form:
x0=xtV
p1V2(11)
t0=txV
p1V2(12)
Since the form of the space-time interval does not depend on whether we
observe a body or not, the resulting form of coordinate transformation de-
pends solely on the relative velocity of the observers, and it does not matter
whether the coordinates of the observers’ systems refer to the observation of
aparticularparticleornot.
It is assumed that the LT is a solution to the space-time interval equation,
and there are many different methods in textbooks to derive this transfor-
mation from the interval equation.
However, in practice, when looking for a solution to the space-time inter-
val equation, one can find an infinite number of solutions that can theoreti-
cally also describe the transformation of coordinate systems. For example, a
pair of equations of the form:
t0=td xe
pd2e2(13)
x0=xd te
pd2e2(14)
satisfies the equation of conservation of space-time interval for any values
of the variables dand ebecause when these values are substituted into the
interval equation, these values are always reduced. Therefore, whatever we
25
substitute for dand e, whether values or functions or even images, these
equations will always satisfy the equation of conservation of the space-time
interval.
And so:
If we substitute d=1and e=V,weobtaintheLorentztransformation.
If we substitute d=cosh and e=sinh ,weobtainasystemofequations:
t0=tcosh xsinh (15)
x0=xcosh tsinh (16)
which are used as one of the methods to supposedly derive the Lorentz trans-
formation from the space-time interval equation, interpreting the solution as
akindof"rotation"inspacetimet, x [57].
Thus, the LT satisfies the principle of conservation of the space-time in-
terval, but it is not its only solution but only one of an infinite number of
such solutions. The fact that the values of dand ein formulas (11,12) can be
arbitrary means that the relative velocity of the bodies (e=V)used in the
Lorentz transformation does not follow in any way from the equation of con-
servation of the space-time interval, as is generally assumed, and is assumed
simply to obtain a solution in the form of the Lorentz transformation.
In summary, in Minkowski spacetime, the LT satisfies the equation of
conservation of the space-time interval. However, it is actually one of an
infinite number of solutions, so within STR, one cannot assume equivalence
between the interval conservation equation and the Lorentz transformation.
This transformation refers only to the relative velocity of the systems of
observers, so in practice, it refers to changes in the geometry of space-time.
3.5.2 Model E4
In the case of the E4 model, we cannot determine any space-time distance
because the dimensions of time and space are not dimensions that make up
reality but only directions interpreted in E4 as space-time dimensions, only
when observing the distinguished body. Thus, to be able to talk about space
dimensions, it is first necessary to determine the observed body because
it is its trajectory in E4 that determines the three orthogonal (precisely
linearly independent) directions perpendicular to this trajectory, and only
these directions are interpreted by all observers of this body as the space
dimensions of their reference frames. This situation has already been shown
in Fig. 11. If we now limit the number of observers from Fig. 11 to only
two, then we have a scheme for observing any distance, but only between the
points of trajectory of the observed body – Fig. 12.
26
Figure 12: Case of two observers x, t and x0,t
0observing the same body
x0,t
0. The angle is a measure of the relative speed of observers x, t and
x0,t
0:V=sin. The angle ↵describes the velocity of the observed body
x0,t
0,relativetotheobserverx0,t
0and ↵+describes the velocity of the
observed body x0,t
0,relativetoobserverx, t.
The transformation of the coordinates of the observer systems results
immediately from the simple geometrical relations of the drawing, and as a
result, we obtain formulas for the transformation of the coordinates in the
form:
t0=t
cosxsin
coscos↵(17)
x0=xtsin
cos↵(18)
And the inverse transformation:
t=t0
cos+x0sin
coscos(↵+)(19)
x=x0+t0sin
cos(↵+)(20)
where
sin=v0(21)
is the relative velocity of the observer x0,t
0in the coordinate system of the
observer x, t,
and
sin↵=v0(22)
is the relative velocity of the observed body x0,t
0in the coordinate system
x0,t
0.
Consequently,
cos=p1V2(23)
27
and
cos↵=p1v02(24)
Now, we can rewrite equations (17)-(20) with the help of the two velocities
(21) and (22), and the transformations take the form:
The new transformation of coordinates:
t0=t
p1V2xV
p1V2p1v02(25)
x0=xtV
p1v02(26)
And the inverse transformation:
t=t0
p1V2+x0V
p1V2(p1v02p1V2v0V)(27)
x=x0+t0V
p1v02p1V2v0V(28)
Note that in such a transformation, we can see that the time dilation
in the observer’s system is identical to the time dilation resulting from the
Lorentz transformation (27), but in the case of space dimensions, we see that
the contraction of length will not occur (28).
In addition, the coordinate transformation does not depend solely on the
relative velocity of the observers, as is the case with the Lorentz transforma-
tion. It also depends on the velocity of the observed body relative to both
observers. Thus, the transformation of the coordinates is not related to a
change in the geometry of space-time but is only the result of the way obser-
vations are carried out along the directions of propagation of interactions in
E4, which are perpendicular to the trajectories of bodies in E4.
In summary, in E4, the coordinate transformation satisfying the space-
time interval conservation equation has a different form than the LT. It pre-
dicts time dilation, does not predict length contraction, and relativistic effects
depend not only on the relative velocity of the observers but also on the ve-
locity of the body observed in both of their coordinate systems. Relativistic
effects are now the result of the way observations are made in E4, rather
than the result of deformation of the geometry of Minkowski spacetime.
3.6 Composing velocities
Since the rule of composing velocities results from the transformation of coor-
dinates, then having two different coordinate transformations resulting from
28
two alternative models, we should get different rules of composing velocities,
and these in turn can give basis to experiments, which can be a test of which
model to choose, or indicate yet another way of describing relativistic effects
than those presented in this article.
3.6.1 Minkowski space-time model
The topic here is generally known. The formula for the composing velocities
resulting from the Lorentz transformation is as follows:
V=v1+v2
1+v1v2
c2
(29)
If, for example, we compose two velocities where v1=v2and plot a graph of
the dependence of the resultant velocity on the sum of these velocities, the
graph looks like this:
Figure 13: Resultant velocity Vas a result of composing two identical com-
ponent velocities v.Onthehorizontalaxis,thevalueofthecomponent
velocities, on the vertical axis, the resultant velocity V.
Of course, the resultant velocity will never exceed the speed of light.
3.6.2 Model E4
In the E4 model, the measure of relative velocity is the angle of inclination of
the trajectory, and the relative velocity equals the sine of the angle between
the trajectories of the bodies. Therefore, the composition of velocities here
involves adding angles between trajectories – this is illustrated in Fig. 14.
29
Figure 14: Composition of velocities in the E4 model. The bodies t1and
t2move relative to the laboratory system and relative to each other. The
measure of relative motion is the angle between the trajectories related to
the observed velocity by formula V=sin'(6). The motion of the body t1
relative to the laboratory system is determined by the angle '1, while the
relative motion of the bodies t1and t2is determined by the angle '2Thus,
the measure of the motion of the body t2relative to the laboratory system
is the angle '1+'2which corresponds to the velocity V=sin('1+'2)
In this case, if the velocity of body No. 1 in a laboratory system is:
v1=sin'1(30)
The velocity of body 2 relative to body 1 is:
v2=sin'2(31)
Then, according to Fig. 14, the resultant velocity of body 2 relative to the
laboratory system should be equal to the sine of the sum of the angles '1+'2
and equals:
V=sin('1+'2)=sin'1cos'2+sin'2cos'1=v1p1v22+v2p1v12
(32)
As seen in the diagram – Fig. 15, the speed of light will not be exceeded,
while the speed interpreted as the speed of light is reached at one of the
points in the diagram after composing two specific values of speed, each of
which is less than the speed of light. A more detailed analysis of the graph
will be presented in the chapter on the proposal of an experiment testing the
behavior of both models in special cases.
30
Figure 15: Result of composing velocities according to model E4. The ve-
locity composing principle does not allow to achieve a velocity greater than
unity (the speed of light), but for certain values of the combined velocities
v,theresultantvelocitycanreachthevalueofV=1, which is impossible in
the case of composing velocities in the Minkowski spacetime model.
3.7 Wave properties of the particle
3.7.1 Minkowski space-time model
In Minkowski’s space-time model, the basic assumption important at this
point is that the dimensions of time and space we observe are the actual
dimensions that make up reality and that all motions are relative and there
is no objective reality in which we can describe all events in a way that is
independent of the observer’s choice of coordinate system.
Although bodies demonstrate wave characteristics under certain condi-
tions, due to the lack of medium and the relativity of the motion of the
bodies, we cannot describe the bodies directly as waves. The equations of
quantum mechanics and wave functions are applied for the description, which
allow us to describe the influence of the wave character of waves on the image
of physical phenomena at short distances. The wave function, which is the
solution of Schrödinger’s equations, takes the form:
=ei
~(Etpr)(33)
In summary, according to the Minkowski spacetime model, bodies move
in relative motion and demonstrate wave properties under certain conditions,
but it is not possible to describe a body directly as a wave.
31
3.7.2 Model E4
In the case of the E4 model, all bodies move along their trajectories at
constant speeds relative to E4, so it is possible to describe the body directly
as a wave. Let me remind you that relative motion has nothing to do with
the absolute motion of the bodies in E4 because relative motion is only a
function of the angle of inclination of the trajectory of the observing bodies
(in E4). It is the relativity of determining the angle of inclination of the
trajectory that speaks about the relativity of the motion, not the very fact
of the motion of the bodies in E4.
The wave function in the form described by formula (33) is a solution
of the Schrödinger equation, which is nonrelativistic, so we treat the energy
and momentum in this function as nonrelativistic values. However, let us
now transform the wave function using relativistic forms of momentum and
energy. Then, we obtain (for a system of units in which c=1):
=ei
~(Etpr)=ei
~(m0dt
ds tm0dr
ds r)=eim0
~(d(t2)
2ds d(r2)
2ds )=eim0
~(d(t2r2)
2ds )=
eim0d(s2)
~2ds s=eim02ds
~2ds s=eim0
~s
(34)
Knowing that
!0=m0
~(35)
and that in E4, the sense of the value of the interval sis limited to the value
of the proper time of the observed body t0,wecanseethatformula(33)
rewritten for the description of the particle in E4 has the form:
=ei!0t0(36)
Or writing the formula differently, we have:
=cos(!0t0)isin(!0t0)(37)
We can see th a t t he wave function, descri b i ng an abstract func t i on in
Minkowski spacetime, which can only be considered using the equations of
quantum mechanics, in E4 is a combination of two flat waves – one with
a real amplitude and the other with an imaginary amplitude. In the E4
model, it is assumed that the imaginary number distinguishes the direction of
motion of the particle in E4, i.e., describes the time-axis of the particle system
(by analogy with the method used to calculate alternating current electrical
circuits, where the values associated with time are defined as imaginary). The
32
other three directions perpendicular to the particle’s trajectory are assumed
to be real.
According to this interpretation, the wave described by equation (37) is a
combination of a longitudinal wave along the direction of motion of the wave
in E4 and a transverse wave in the direction perpendicular to this direction of
motion (the time-axis). In other words, it is a vortex in the plane determined
by the dimension of time of the wave and the direction perpendicular to this
dimension of time. The use of complex notation does not mean that the E4
space is a complex space. The imaginary number distinguishes the direction
in E4 interpreted as the time dimension of the particle, denoted as imaginary,
from the three remaining directions perpendicular to the time-axis of the
observed body, marked as real and interpreted by the observer observing
this body as its space-dimensions of his reference frame x, y, z.
If we wanted to describe the particle as a wave in E4, then to obtain the
finite energy of the particle, it would be necessary to put on it the amplitude
having maximum at the center of the wave and decreasing with the distance.
The shape of the amplitude is not yet determined. For now, we assume an
exemplary amplitude of A(r). The wave function describing the particle now
has the form:
=A(r)[cos(!0t0)isin(!0t0)] (38)
Equation (38) describes a particle in a system bound with the particle moving
along the particle’s trajectory with velocity V=1.Sinceitisanabsolute
motion and the trajectories of the particles are absolute, we can also describe
the particle in the absolute coordinate system bound with E4. To do so, we
need to add a factor to describe the motion of the particle along its trajectory
in E4. The function now takes the form of:
=A(r){cos(!0t0)i[t0+sin(!0t0)]}(39)
We know t hat the imaginar y p a r t of the function de s c rib e s t he longitudina l
oscillations along the direction of motion of the particle in E4 interpreted as
the time-axis of the body t0. The real part describes the oscillations along one
of the other x, y, z directions interpreted by the observer of this body as the
space dimensions of the observer’s coordinate system. All these dimensions
x, y, z lie in a subspace perpendicular to the trajectory of the particle - t0
-in E4. Let us assume that the real part of the function (39) describes the
oscillations in the direction interpreted by the observer as the xcoordinate.
In this case, the distance ris the distance measured from the trajectory of the
particle in the plane perpendicular to the trajectory of the particle formed
from the two remaining dimensions: y, z in the observer’s coordinate system,
33
so formula (39) can now be written as:
=A(py2+z2){cos(!0t0)i[t0+sin(!0t0)]}(40)
Since our formula describes the shape of a four-dimensional particle, we need
to zero one of the dimensions to plot its shape. Let us assume that the
dimension z=0;then,theshapeofourwavecanberepresentedasin
Fig. 16. The direction r here is directed along the y-axis of the observer’s
coordinate system.
Figure 16: A function that describes a wave as a deformation of space ac-
cording to formula (39). An exemplary amplitude shape is assumed. A wave,
as a combination of longitudinal and transverse oscillations, is in practice a
vortex – in the figure it is a vortex in the t, x plane. The vortex-wave travels
along the t0axis in E4 with an absolute velocity of V=1.
Note that the real part of the particle in formula (39) describes the ridge
of the wave, i.e., the directions perpendicular to the trajectory of the particle
in E4. Since the interactions propagate perpendicular to the trajectory of
the particle, it follows that they propagate along the ridge of the wave. To
enable such propagation of interactions, let us expand the equation of our
particle by adding the phase velocity of the wave along the ridge of the waves.
In this case, it is logical to assume that this phase velocity is equal to the
speed of wave propagation in E4, i.e., Vphase =±1.
Thus, the final equation of such a wave/particle presented in the absolute
coordinate system in E4 has the form:
=A(r){cos!0(t0⌥r)i[t0+sin!0(t0⌥r)]}(41)
34
To repres e nt the shape of the fun c t i on graphically, anal o g ously to formu l a
(40), we assume that the real part of formula (41) determines the oscillations
along the xdirection, i.e., We write function (42) analogously to formula
(40), and we have:
=A(py2+z2){cos!0(t0⌥r)i[t0+sin!0(t0⌥r)]}(42)
If we now zero the z coordinate, the shape of our function will be as shown
in Fig. 17.
Figure 17: The wave described by equation (41,42) is represented in t0,x,y
coordinates. The wave is now a vortex moving in E4 with an absolute velocity
of V=1along its trajectory (time-axis) of the wave. At the same time, the
phase at the ridge of the wave moves toward-, and outwards- the center of the
wave with velocity Vphas =±1.Anydisturbanceofthewavepropagatesin
E4 along the ridge of the wave with the absolute velocity, being a composition
of the velocities Vphas =±1and V=1.
We can see that now the wave, which in Fig. 17 is represented in the
space E4, makes a rotational motion in the plane t0,x.Inturn,ifinformula
(42) we zero the real part corresponding to the direction perpendicular to
the trajectory interpreted by the observer as dimension x,thenwecanshow
what the ridge of the wave looks like in the subspace t0,y,z, which is now
described by the equation:
=iA(py2+z2)[t0+sin!0(t0⌥r)] (43)
The shape of the ridge of the wave for Re( ) = 0 is shown in Fig. 18a, which
shows two such wave exchange interactions.
35
Figure 18: Observation mechanism. Fig. a shows the ridges of the wave while
exchanging interactions, the directions of information transmission along the
ridges of the waves and the actual path of the signal in E4 (between the ABC
points) resulting from the combination of the motion of the waves in E4, and
of the motion of disturbances along the ridges of these waves. Fig. b. It
shows the segments that the signal travels (along the trajectory of the waves
and their ridges) – these are two segments a and two segments b. In Fig. c,
the directions interpreted by the observer as axes of his coordinate system
x, t during the observation of body t2are presented. The CA segment is the
path that the observer thinks the signal takes, and the OA segment is the
direction interpreted by the observer as the spatial axis (spatial dimension)
of his coordinate system. In the case where the trajectories have a common
point where the bodies have met or will meet, segments aand b(Fig. b.) are
equal to each other, and the angle between segment OA (space-axis of the
observer) and the trajectory of the observed body t2is a right angle (Fig. c.).
Therefore, to represent one four-dimensional wave, we need to use two
figures – 17 and 18a – showing the intersection of a four-dimensional wave
with a plane. In Fig. 17, the plane z=0,andinFig. 18a,theintersection
of the same wave with the plane x=0.
The direction of phase motion along the wave ridge depends on the sign in
formula (41) for the phase velocity. We can see that the particle described by
formula (41) rotates in a specific plane (here t0,x)andhasaspecificdirection
of the phase – Fig. 17.
Now, if our particle is to receive and send disturbances along its ridge,
then its basic component should be not a single wave but a combination of
two waves described by equation (41) – one with the phase velocity sign plus
and the other with the minus sign. As a result, to represent one component
of a particle, we need a combination of two such fundamental oscillations.
Since the rotations of such a particle take place only in one plane – in
36
Figs. 17 and 18a it is the t0,xplane, so to describe a particle having identical
properties in all three space-dimensions of the observer’s system, it is neces-
sary to combine three pairs of such oscillating particles – or rather rotating
not only in the t0,x plane but also in t0,y and t0,z ones.
Thus, the fundamental particle should consist of six such fundamental
oscillations described by formulas analogous to formula (41).
Now that we have an idea of the approximate wave shape of the particles,
we can now describe the process of exchanging information between bodies.
Interactions/signals are transmitted by means of disturbances propagating
along the ridges of the waves with phase velocities in the system of each
particle/wave. The process of transmitting interactions/signals is shown in
Fig. 18.
Fig. 18a shows the ridges of the two waves. Waves move along trajectories
that have a common point. The angle between trajectories is a measure of
the relative velocity of waves/bodies according to formula (6).
Awavetravelingalongatrajectoryt1is an observer wave, i.e., it receives a
signal.
Awavetravelingalongtrajectoryt2is an observed body, i.e., it sends a
signal.
Fig. 18a shows the path of the signal propagating along the ridges of the
waves. This are the ABC segments.
The observed body t2sends a signal at point A. The signal travels along
the ridge of the wave with a velocity of Vphas =1, and together with the
wave t2, it moves along the trajectory with a velocity of V=1. Thus, the
signal travels on a trajectory inclined at an angle of 450to the trajectory of
the observed body t2.
At point B, the signal catches up with the ridge of the observer’s wave
t1,movestotheridgeoftheobserver’swave,andfromthattimepropagates
along the ridge of the wave receiving the signal (observer), now with a phase
velocity of Vphas =1, and together with the wave along trajectory t1with
avelocityofV=1. Eventually, the signal travels now at an angle of 450,
but already to the trajectory t1of the body - the observer.
The distance traveled along the ridges of both waves by the signal is shown
in Fig. 18b.
The distance traveled along the ridge of the sending wave is equal to a,
and the same is true for the distance traveled by the sending wave along its
trajectory.
The same goes for the receiving wave.
The signal travels along the ridge of this wave at a distance band is the same
along the trajectory of the wave receiving this signal.
Thus, the method of transmitting information treated as disturbances
37
moving along the ridges of waves with a phase velocity equal to unity causes
that regardless of the angle between the trajectories of the observer and the
observed body, i.e., the relative velocity of the observers, the path traveled
by the signal measured along the trajectory of the bodies from the moment
of sending to receiving the signal is:
t=a+b(44)
and is equal to the path of the signal traveled by this signal along the ridges
of the waves.
s=a+b(45)
As a result, the speed of signal propagation is constant and equal to unity
regardless of the relative velocity of the bodies (i.e., the angle between their
trajectories) – Fig. 18b
V=s
t=1 (46)
The observer does not observe the trajectory of the signal. The true
trajectory of the signal shown in Figs. 18a and 18b, represented by segments
AB and BC,isnotavailableforobservation.
The observer can only know point Awhere the signal was sent and point C
where the signal was received. Hence, the observer has the impression that
the path traveled by the signal is segment AC –Fig.18c.
Given the path AC, own trajectory, and the information that the path
traveled by the light along the space-direction is equal to the time elapsed
from the moment of sending to receiving, and this time is equal to the segment
OC,theobservercandeterminethesegmentOA (which must be equal to
the segment OC), which for him is the space-axis of his coordinate system.
It is a simple geometrical task. We have to plot an equilateral triangle
with base CA (Fig. 18c) and one of the sides pointing along the trajectory
of observer t1. The solution relies on plotting the height of the equilateral
triangle perpendicular to the base CA. The intersection of the height of the
triangle and the trajectory of the observer gives point O–thevertexofthe
equilateral triangle. Now, the OC segment describes the time measured in
the observer’s system from the moment the signal is sent from point Ato the
moment of receiving at point C, while the OA segment with a length equal
to the OC segment is interpreted by the observer as a spatial distance that
determines the direction interpreted by the observer as the space direction.
Therefore, the observer also registers the speed of propagation of signals equal
to V=OA/OC =1.
Here, we have a very interesting property. Only in the case where the
trajectories have a common point where the waves have met or will meet in
38
the future does the direction interpreted by the observer as the space dimen-
sion of OA coincide with the ridge surface of the wave sending the signal
(observed), i.e., the space direction is perpendicular to the trajectory of the
observed body (sending the signal). However, the case of motion along tra-
jectories having a common point is a special case of motion. In addition,
what about particles moving along trajectories that do not have such a com-
mon point, i.e., those for which the OA segment is not perpendicular to the
trajectory of the observed body?
The relativistic equations for such particles, such as the equation deter-
mining the time dilation (7,8), will have a different shape than the known
relativistic relationships. Is the special case the norm then? Note that the
Big Bang theory assumes that all particles move along trajectories having a
common origin. Thus, the "special case" becomes the norm, and the rela-
tivistic formulas, as we know them, become the evidence of the Big Bang.
However, if in distant regions of the Universe as a result of the expansion
of the Universe and various kinds of motions of matter, there have been sig-
nificant deviations from the motion along the original trajectories, then we
should sooner or later observe some deviations from the known relativistic
principles.
Thus, within the E4 model, starting from the wave function being
the solution to the Schrödinger equation, we came to the justifica-
tion of the constancy of the speed of light and proved the existence
of the Big Bang. Other consequences of this fact, such as Hubble’s law,
will be discussed in the following chapters.
In addition, the model of a body described as a wave allows for a natural
definition of the particle’s own clock, which can simply be the number of
oscillations of the wave that makes up the particle. More on this subject in
the papers [56,58].
At the end of this subsection, it is worth interjecting an im-
portant piece of information.Letusdrawattentiontothefactthatin
Fig. 18.c for the observer t1the points Oand A(lying on its spatial axis)
are simultaneous. However, the condition that the trajectories of the bodies
must have a common point where they meet or will meet, combined with the
assumption that the velocity of the bodies along their trajectories is identi-
cal, says that in E4, simultaneous are the points on the bodies’ trajectories
that are equally distant from the common point of the trajectory. Thus,
points Oand Aare not simultaneous in E4. The problem of simultaneity is
described in detail in the paper [8]. In this paper, I present bodies moving
along trajectories having a common origin. Such a presentation of the obser-
vation problem is not necessary, but a different presentation of the problems
would require an additional discussion of the simultaneity problems, which I
39
wanted to avoid in order not to overcomplicate the paper, whose main pur-
pose is to present the most important mechanisms of the proposed approach.
The common origin of the trajectories simply clearly defines the simultane-
ity problem in E4 and this is the only reason why such a presentation of the
problems is used in the figures.
To sum up: In E4, bodies can be described directly as waves that are
oscillations of space. The waves are moving in E4. In the case of rectilinear
trajectories, the speed of the waves is constant and equal to unity. Distur-
bances of the waves travel along the ridges of these waves with the velocity
Vphas =1, which justifies the mechanism of the phenomenon of constancy of
the speed of light and the fact of interpreting directions perpendicular to the
trajectory of the body in E4 as space-dimensions of the observer’s coordinate
system. At the same time, the number of periods of such a wave can play a
role in the particle’s own clock indications. An additional conclusion can be
the proof of the existence of the Big Bang.
3.8 Conditions for observing the bodies around us
3.8.1 Minkowski space-time model
In this case, the matter is obvious. Interactions are carried by quanta, which
are independent particles propagating in space-time along spatial dimensions
at the speed of light, regardless of the motion of bodies: sending and receiv-
ing the quanta. According to Einstein’s first postulate, phenomena do not
depend on velocity relative to the observer. On the other hand, the constancy
of the speed of light is ensured by Einstein’s second postulate.
This is basically the main mechanism of our communication with the
world around us proposed by the theory of relativity.
3.8.2 Model E4
In the case of the E4 model, the construction of reality itself is very simple,
while the observed image transmitted between bodies.
The exchange of information proposed in the E4 model is discussed in the
previous chapter. It involves the exchange of disturbances between particles
treated as waves. With such a representation of particles, the disturbance
moves along the ridges of the waves – the observer and the observed particle.
Initially, the disturbance moves along the ridge of the wave sending the signal
and then reaches the ridge of the receiving wave, and it moves along the ridge
of the wave receiving the signal, i.e., the observer (Fig. 18).
As already mentioned in the chapter "The two reality models conserving
40
the space-time interval", there are two types of directions associated with
each body in E4, which are constant for a given body and do not depend on
the observer’s choice. These are – the direction of motion of the body in E4
interpreted as the time-axis of the coordinate system of this body – t0–and
the three directions perpendicular to the time-axis of this body, which all
observers of this body interpret as the space-axes of their reference frames.
As we already know, these three directions are determined by the ridges of
the waves describing the bodies.
However, due to the motion of the body in E4 at absolute velocity V=1
along its trajectory in E4 and the motion of the disturbance along the ridge
of the particle at the phase speed of the wave Vphas =±1,therearealso
directions of signal propagation in E4. We denote them as Vs–thedirection
of the signal sent by the observed body and Vr–thedirectionofthesignal
received in the observer’s system. These directions are also constant in E4,
and their trajectories are inclined at an angle of 450to the time-axis (trajec-
tory) of each body. As absolute directions, they also do not depend on the
choice of the observer’s system. The speed of signal propagation in E4 is a
combination of two mutually perpendicular velocities: Vphas =±1and V=1
, so the speed of signal propagation in E4 is constant and is Vs=Vr=p2.
All these absolute directions bound with the particle are shown in Fig. 19.
41
a b c
Figure 19: The absolute directions bound with the particle in E4 are pre-
sented for the case of observing bodies moving along trajectories having a
common origin. All directions are resented in Fig. a. In Fig. b are shown di-
rections that are important for sending signals, i.e., for the observed particle.
In Fig. c are shown directions that are important for the particle receiving
the signals – the observer. V-itisthespeedofwavepropagationalongthe
particle’s trajectory in E4; V=1.Vphas – it is phase velocity, with which
disturbances propagate along the wave ridge; Vphas =±1.Vr,Vsthe are the
speeds of the signals sent and received. The signals travel on trajectories in-
clined at an angle of ±450to the trajectory of the body sending/receiving
the signal. The signals travel at an absolute speed equal to p2.Signalsare
disturbances related to the motion of the wave-bodies in E4.
The exchange of information proposed in the E4 model is based on the
exchange of disturbances between particles treated as waves. According to
the E4 model, disturbances, treated as quanta, are not separate particles (as
is currently assumed), but they are disturbances related to the wave structure
of the particles.
Since the observer is moving along its trajectory at absolute speed V=1,
for the signal sent by the observed body to be registered by the observer, this
signal must first catch up with the ridge of the observer’s wave and then move
along the ridge of the observer’s wave. For this to be possible, the projection
of the signal’s velocity onto the time-axis (the observer’s trajectory) V0
smust
be greater than one, or in the case of parallel trajectories, equal to one. This
situation is illustrated in Fig. 20.
42
Figure 20: The observed body t0sends a signal that propagates at the speed
of Vsignal =p2. The signal moves relative to trajectory t0at an angle of 450.
The body t– the observer – moves with velocity V=1.Forthesignaltobe
received by the t-body, it must catch up with it. Therefore, it is necessary
that the projection of the signal velocity (V0
signal)onthetrajectoryofbodyt
is greater than 1. When the angle '=90
0is exceeded, the projection of the
velocity of the signal on trajectory twill be less than one, i.e., the signal will
never catch up with the observer’s wave ridge, and it will not be possible to
register it by the observer. The figure shows a case where the signal catches
up with the ridge of body/wave tand then propagates along the ridge of this
wave at an angle of 450to the trajectory of observer t.
Thus, if the trajectories of the observed body and the observer are inclined
to each other at an angle ',thentheconditionofobservationofthebody
by the observer can be written as:
V0
signal =Vsignal cos('450)=p2cos('450)1(47)
Aprojectionofthesignal’svelocityontotheobserver’stime-axisV0
signal
with a value of <1means that the signal will never "catch up" with the
observer’s wave-ridge, so simply observing such a body is not possible. It is
easy to see that for the case in Fig. 20, the angle between the trajectories
between 900and 2700describes the trajectories of bodies that cannot be
43
observed with quanta and probably do not interact with surrounding bodies
in the way we know from our observable World. Examples of experiments
using this mechanism will be discussed later in this paper.
The problem of signal propagation is described in more detail in [56,58].
Animations showing the way the disturbance propagates –, i.e., the method
of mutual observation in the described case – are available in presentations
at YouTube.com [58].
3.9 Chapter Summary
The main assumptions of the alternative model of reality are presented, and
the meaning of basic concepts and the description of basic phenomena are
compared with analogous ones in the Minkowski space-time model. In addi-
tion, the method of observation proposed in the E4 model, different from the
method of observation adopted in the Minkowski spacetime model, predicts
the occurrence of various effects not predicted by previous models.
If the new approach is accepted, the scope of the resulting changes will
be much wider than it results from the above descriptions. The adoption of
the E4 model eliminates the need for covariant notation. On the other hand,
interpreting bodies directly as waves will most likely result in a departure
from such concepts as field, mass or electric charge and will eliminate the need
to use the current tools of quantum mechanics. This will cause replacing the
equations of field theory or quantum mechanics with the simple interaction
of waves treated as deformations