Proposal of simpler description of SRT
W.Nawrot
A new concept of four dimensional reality is
presented. The fourth dimension of the reality is now described with a
dimension different from the time of the observer. Consequently, the Euclidean
model of reality is obtained, description of phenomena is simplified in
relation to the four-dimensional Lorentzian space-time and the singularities
taking place in the description of the reality
become now an effect of performing the observation and are not the
property of reality any more. The new model also predicts certain new
experimental effects which can be a reliable test for the new model.
Introduction
The question in what reality we live
has been absorbing our minds for centuries. When we observe and investigate the
surroundings, we only perceive certain events taking place in the reality,
whereas the reality itself cannot be observed. On the basis of the observation
of events, we build models which enable us to picture the reality to ourselves
with the help of the concepts we can comprehend.
Construction of a model of reality,
like the construction of a building, should be based on solid foundations. Accepting a certain seemingly obvious
assumption, which has been never checked, as a basis for the model, may result
in further complication of the theory or lead to some false conclusions.
In this paper I would like to point
out one such assumption seemed obvious enough
not to be discussed till now. Namely, the definition of the fourth dimension,
creating the space-time.
The similarity of time- and space
dimensions in relativistic formulas suggests that the reality is
four-dimensional and the fourth dimension should be described with time. It
seems that time should be taken directly as the fourth dimension that creates
the reality. However, the statement that time is the fourth dimension is
equivalent to the assumption that in the four-dimensional reality, the time
dimension has to be perpendicular to the three space dimensions. But does it
really have to?
Theoretically, in order to describe
the fourth dimension with the help of time, it is not necessary to assume that
time is perpendicular to the three space dimensions. It is enough to state that
it is aligned to them at any non-zero angle.
The assignment of perpendicularity of the time
has been accepted without any proof – only by analogy with the three space
dimensions. Is this assumption correct? I will try to answer this question,
describing a hypothetical reality, in which the time dimension is not
perpendicular to the three space dimensions. As a result we will obtain a
description of events simpler than the description resulting from SRT; however,
we will also obtain some new, different conclusions which can be experimentally
verified. Those conclusions should give
us a final answer to the above question.
How does
the new model work?
The coordinates of time and space of
the frame of any observer must satisfy the rule of the conservation of the
space-time interval, which in the case of observation of a particular body by
many observers takes the following form:
(1) dt’2=
dti2- dxi2- dyi2-
dzi2
where dti, dxi, dyi,
dzi, denote the space-time coordinates in the frame of the i-th
observer, and dt’ denotes the proper
time in the frame of a body in motion.
Up
to now it was assumed that the four components of the equation (1): dti,
dxi, dyi, dzi, describe the dimensions of the
reality (are perpendicular to one another), while dt’ is the distance in this
reality. We call the four-dimensional
reality in which the distance is measured according to equation (1) the
Lorentzian space-time. However, the equation (1) allows also to describe a
different reality. Let us write the equation (1) in the following form:
(2) dti2= dt’2+dxi2+ dyi2+ dzi2
If the values dt’, dxi,
dyi, dzi will be taken as the coordinates describing the
reality- the dimension described by the coordinate t’ is now perpendicular to
the dimensions - x,y,z –, we will obtain the Four-dimensional Euclidean
Reality - FER. In FER, the fourth
dimension is described with the proper time of observed body, whereas the time of
the observer dti becomes now the distance. The dimensions of both realities, i.e.
t,x,y,z and t’,x,y,z, satisfy the
equations (1) and (2); however, accepting the proper time of the observed body
as a value describing the fourth dimension results in a reality with different
properties from the one we have known till now.
The differences between Lorentzian
space-time and FER were shown for two-dimensional case in the fig. 1.
Fig.1. The difference
between the Lorentzian space-time and FER in case of the observation of a
single body:
a.) in the Lorentzian space-time, the rule of
measuring of the distances – formula (1) – forces the deformation of dimensions (the time
dimension of the observed body is stretched), but the observed dimensions are
also the dimensions which create the reality. The time dimension of the
observer’s frame is here perpendicular to the space dimensions.
b.) in FER the dimensions are not deformed.
This is possible at the cost of an assumption that the reality is constructed
of dimensions other than the observed time and space. The time dimension of the
observer is not perpendicular to the space dimensions any more, as it takes
place in the Lorentzian space-time – fig. a). The dimension perpendicular to
the space dimensions is now described by the proper time of the observed body.
In both cases shown in
this picture Dt=5, Dx=4, Dt’=3 (in arbitrary units).
Accepting the FER model, describing the reality
with the coordinates t’,x,y,z, means that, contrary to the Lorentzian space-time model,
- Dimensions
which we are able to observe are not the dimensions which create the
reality any more. While the space
dimensions describe „true” distances in
FER, the time dimension of the observer’s frame does not denote the
distance along the fourth dimension of the reality. The fourth dimension
is now described with the proper time of the observed body, whereas the
value we perceive as the time is a composition of space dimensions and the
„true” fourth dimension. The problem of time is discussed in more detail
in the next paper [1].
- FER
is the Euclidean space so, contrary to the hitherto Lorentzian space-time,
its dimensions are not deformed.
If the dimensions of FER cannot be
deformed, then how can we ensure the conservation of the space-time interval?
This question is explained in fig. 2
where we are shown (in FER) the observation of one body by several observers
(it is described with equation 2). In
FER the space dimensions of the observers’ frames are perpendicular to the time
axis of the observed body, so the space axes of all observers’ frames of the
same body overlap(x1=x2).
The time axes of the observers (t1and
t2) are now inclined to the
time axis of the observed body at angles ji (i=1,2), which – as we can see in
the fig. 2 – denotes the velocities of the body in relation to the observers,
according to the formula:
(3) 
Fig.2 Observation
of the body by the observers: “1” (coordinates system x1t1)
and “2” (coordinates system x2t2) , which
move in relation to the body with different velocities. The observer “2”
is moving faster than “1”. The observation is expressed in the FER coordinates
system x,t’.
It can be seen that, according to
the above conditions, the rule of conservation of the space-time interval -
formula (1) - is always satisfied, independently of the choice of the
observer’s frame. At the same time all observers can see that if the time Dt’ passed in the frame of the observed body, then
the times equal to Dti passed in their frames. It results
from the fig. 2 and formula (3) that those times are connected to one another
with the following relation:
(4) 
The time dilation in the moving
frame, measured by the observer, is not the result of the deformation of
dimensions in FER – as it was assumed till now in the Lorentzian space-time. It
is only the result of inclination of the time axis of the observer’s frame to
the axis of time of the observed bodies frame. The angle of the inclination
depends on the relative velocity (3).
Dimensions of all the bodies’ frames
are expressed in FER in the same scale, independently of the relative motion of
these bodies. According to the rule shown in the fig. 2, it is also possible to
derive the Lorentz Transformation in a very simple way [2].
If we now choose a different body as
the observed one, then the space axes of the observers’ frames have to be
chosen as perpendicular to the axis of time of this observed body’s frame. It means
that in FER there are no directions assigned a priori as the space or time. We perceive three directions of the
four-dimensional reality (FER) as the space dimensions, and one as the time
dimension. These directions are
determined individually for every process of observation and specifically for
every couple: observer – observed body. The same direction in FER can be
interpreted during one observation as the time dimension and during another
observation as the space dimension.
Finally we can compare the
properties of FER and the Lorentzian space-time:
|
FER |
Lorentzian space-time |
|
The reality is constructed of four
identical dimensions, and none of these dimensions can be assigned in advance
as the space- or the time one. |
The reality is constructed of four
dimensions. One of them denotes the time; the other three denote the space
distances. |
|
It is only the choice of the
observer and the observed body that determines which direction of the FER is
interpreted as the time- and which is interpreted as the space-dimension. |
The directions of the space-time
being the time- and the space-dimensions are assigned in advance and do not
depend on the choice of the observer or of the observed body. |
|
The direction perpendicular to the
three directions, interpreted as the space dimensions, is described with the
proper time of the observed body. The time of an observer is the distance in
FER. |
The fourth dimension,
perpendicular to the three space dimensions, is described with the time of
the observer. In the case of observation of a body (the size of which can be
neglected), its proper time has the meaning of the distance in space-time. |
|
The dimensions of bodies in motion
are not deformed. Directions interpreted as the time- or space-dimensions are
chosen individually for every process of observation and they are inclined to
one another at an angle depending on the relative velocity. They are
therefore observed as if they were deformed – see fig.2, formula (4) |
The dimensions of bodies in motion
are deformed and this deformation causes the time dilation and the length
contraction. |
|
Distance
in the four-dimensional reality is defined identically as in the three-dimensional
space – see formula (2). The distance cannot be equal to zero while if any of its components on the right side of
formula (2) is a non-zero value. The singularities similar to those taking
place in Relativity Theory (see right) do not appear here. |
Distance in the four-dimensional
space-time is defined according to different rules than in the three
dimensional space – see formula (1). The distance can be equal to zero while
the space and time distances are non-zero values. This is the cause of the
singularities, which occur in the Relativity Theory description |
Table 1. Comparison of properties of FER and the Lorentzian space-time
We only know the properties which
FER should have. Now we could ask the
question: How does the reality, which fulfills
the above properties, look like, and why is it observed as the
Lorentzian space-time?
The construction of FER
According to the previous
considerations we accept that:
Assumption 1: Reality is the four-dimensional Euclidean space. None of
the dimensions of this space has the meaning of the time- or the
space-dimension assigned in advance.
Assumption 2: In the four dimensional reality there exist bodies. The
bodies move along certain
trajectories, and all trajectories are allowed. Length of the trajectory passed
by the body is a measure of its proper time.
If we use the concept motion
along trajectory, then we should relate this motion to a certain time.
In FER none of the dimensions has the meaning of time, so in order to define
the motion in FER we have to introduce an additional value - THE
SUPERTIME. THE SUPERTIME is not the
fifth dimension in the sense of the four dimensions creating FER. It is a
parameter, which allows for putting in order all the events along the
trajectory of the body. A detailed definition of THE SUPERTIME is presented in
the next paper [1]. Since the length of the trajectory passed by the body is
always somehow proportional to THE SUPERTIME, then in this paper I will only
assume that the length of the trajectory is the measure of the proper time of
the body. The subject of this paper is a
new definition of the coordinates system describing the reality, so the wave
structure of particles – described also in the next paper [1] - will be disregarded for the time being, and
the particles will be treated here as points.
If the length of the trajectory of
the body is a measure of its proper time, then the trajectory of this body
should be the time axis of its coordinates system. Then, according to the
previous considerations, we should accept the following:
Assumption 3: The directions in FER, perpendicular to the trajectory of
an observed body, are interpreted as the space-dimensions.
We can now interpret the meaning of
particular components shown in the fig. 2.
· The time axes t’, t1 i t2 now denote the trajectories of
bodies in FER.
· Angles between trajectories in FER
denote the relative velocities, which the observers measure in their frames xi,ti.
· The space axes of all observers were
chosen as perpendicular to the trajectory of the observed body t’, and it means that only this body can be actually
observed. If we would like to show, in fig. 2, the observation of a different
body, for example body t2, then we would have to change the
space axes of all bodies into perpendicular to the trajectory of this body,
i.e. to the axis t2.
Since during observations of various
bodies we interpret different directions in FER as the space dimensions, it is
not possible to show the observation of many bodies in the frame of a single
observer (at the same moment of time). In FER the coordinates system of the
observer must be defined from the beginning for the observation of every single
body. Therefore, the complicated picture of the reality which we observe comes
from the composition of single observations performed in different moments of
time (more accurately: in one point and in one moment of time we can register
only one event or total effect of superposition of a few events). From many
separate observations we get the picture of reality similarly as the picture on
a TV screen, where, although each pixel of the screen lights at a different
time, we can see the whole picture as if all the pixels lighted simultaneously.
Because the space dimensions create three dimensional subspace in FER, every
single event is observed as if it took place in the three-dimensional space.
The picture which we receive as a result of composition of many single
observations makes the impression that we live in the three-dimensional
reality. If we add the time (the length of the trajectory passed by us in FER)
to the three observed space-dimensions, we will get the well-known Lorentzian
space-time. However, it is not the “true” picture of the reality, but only its
“projection”, or, in fact, the composition of a set of “projections”,
corresponding to the observations of single events.
The observations are performed with
help of quantum. In order to justify the observed constancy of the speed of
light independently of the observer’s motion, it is necessary to accept the
following assumptions:
Assumption 4: The trajectory* of quantum is perpendicular to the
trajectory of the body which emitted this quantum.
All points along the space axis of
the observer (perpendicular to the trajectory of the observed body) correspond
to the same moment of time in the observer’s frame. In order to register quantum in the
observer’s frame in different moments of time, it is necessary to accept the
next assumption:
Assumption 5: The trajectory* of
quantum is carried along the trajectory of the observer.
The trajectories* of
quantum in FER, which are the result of assumption 4 and 5, are shown in the
fig 3. The trajectory of the observed body (emitting quantum) is shown there,
as well as the trajectories of two observers moving in relation to this body
with various velocities. It is visible that, when the quantum moves*
along its trajectory* by Dxi , its trajectory is carried*
along the trajectory of the body receiving quantum by Dti. This effect does not depend on the
angle of inclination of the trajectory, i.e. on the relative velocity of the
observer. Therefore, the observed velocity of quantum, equal to Dx/Dt, is always
constant and does not depend on the velocity of the observer in relation to the
source of the radiation. In FER, where time and space dimensions are expressed
in the same scale, we assume that:
Assumption 6: In vacuum and in absence of gravitational field Dxi=Dti
so the velocity of propagation of quantum in empty
and not curved space is equal to 1. We have to take
such an assumption because the velocity of quantum in non-curved vacuum is
equal to the maximal velocity of bodies resulting from formula (3). Since the speed of propagation of the interactions
results, in FER, from a different mechanism - assumption 6 - than
the limitation of the velocity of bodies
– formula (3) – then in a medium different from the vacuum
or in the presence of a gravitational field those values may differ. .
Let us notice that, according to the
assumptions 4 and 5, the resultant trajectory of quantum in FER will be
different for every observer. The quantum hitting the body A are carried along the trajectory
of the body A, the quantum hitting the body B are carried along the trajectory
of the body B etc.
Fig. 3. Two observers
watching one body with the help of quantum. The resultant trajectory of quantum is a
composition of motion along the trajectory perpendicular to the observed
body (the space axis of the observers) and carrying the trajectory along the
trajectory of the observer. In vacuum and in the absence of gravitational
field: Dx1=Dx2=Dt1=Dt2
It means that already in the moment of
emission, the quantum must “know” by which body it will be received.
Hence:
1. The emission of quantum must be a result
of a certain interaction between two specific particles. The character of this
interaction is not known yet. It may be, for instance, a resonance of particles
proposed in [3]. The quantum cannot be therefore emitted
somewhere into empty space and move like a particle until it reaches
any random body and is absorbed by it.
2. The idea of trajectory and motion of
quantum must be a conventional notion, because we are only able to know the
points of emission and absorption of the quantum. We are not able to examine the route of the
quantum or what happens with it between the emission and absorption,
because the quantum can interact only with the body towards which it has been
sent.
3. All particles of the Universe must
be somehow informed about the existence of other particles, because already in
the moment of the quantum’s emission, the place and the particle which the
quantum will hit is well-known, apart from the time needed for it. It is,
somehow, a different formulation of the Mach principle, and it confirms the
suggestions about the wave structure of matter proposed in [3] and [1]
The picture of reality proposed in
this paper is very different from and has different properties than the
well-known Lorentzian space-time. If this
is the case then in addition to the description of events similar to those once
proposed by SRT, we should also get a description of some new events which are
not predicted by the Relativity Theory. Examination of these new events should
be a reliable test of the correctness of the FER theory proposed here.
What new phenomena result from the FER model?
The new definition of velocity as
the angle between trajectories of bodies in FER changes the rule of the
composition of velocities. In FER, the composition of velocities consists in
the summation of angles between the trajectories. The rule of the summation of
angles is shown in the fig. 4.
Fig.4 Composition of velocities according to the FER:
·
The moving frame moves in
relation to the observer in rest
with velocity V=sina;
·
The body A moves in relation
to the moving frame with velocity v =sinb;
·
The body A moves in relation to the
observer in rest with velocity Vres=sing=sin(a+b)
As a result of such a summation we
can obtain, for instance, the trajectory perpendicular to the trajectory of an
observer. According to the formula (3), it is equivalent to the acceleration of
the body to the speed of light; according to the SRT-model it is impossible.
Further acceleration results probably in the inversion of the time flow in the
particle’s frame. The attempt on discussion
of this problem is presented in the next paper [1]. One of the still unsolved problems is whether
and how can we observe those particles in FER. We can expect that the new rule
of composition of the velocities should be observable in the case of the
spontaneous decay of relativistic particles.
In such case, products of the decay should depend on the velocity of the
particle. For example, for a strictly
determined velocity, one of the products of decay of such a particle would be a
particle moving with the speed of light (see fig. 4)
Another
phenomenon predicted in the FER, which greatly changes our idea of reality, is
the recession of galaxies. According to today’s knowledge
the running away galaxies are still accelerating –
the velocity of the galaxies is approximately proportional to the distance from
the galaxies. In order to explain the acceleration of the galaxies, the idea of
an additional hypothetical repulsion field propelling those galaxies has
been introduced.
However, according to the FER model,
if the trajectories of galaxies have approximately common origin, then the
observer must observe from his own trajectory that the velocities of galaxies
are proportional to the distances, and it has nothing to do with the acceleration [4]. It results directly from the fact that in FER
the space dimension is chosen individually for the observation of every galaxy
as perpendicular to the observed galaxy’s trajectory. All galaxies are actually
moving along their trajectories in the same way from the beginning of the
Universe, and the observed acceleration is only the seeming effect which
results from the way we observe the reality. Moreover, another immediate result
of the FER model is that the Hubble constant is equal to the inversion of the
age of the Universe and decreases with time; the more detailed description of
this problem is described in [4]. Hence, the FER model allows to simplify the
description of the reality by eliminating the need of introducing any odd
repulsion fields or complicated cosmological models, which were supposed to
explain the meaning of the Hubble’s constant. In FER, all those problems are a
simple consequence of the manner of the performing of observations, and the
derivation of all the above-mentioned conclusions only takes one line of text
[4]. The following test for the correctness of the presented theory results
from the solution of the recession of galaxies presented in FER:
If we assume the theory
of Big-Bang as true, we can expect that the trajectories of galaxies fill
almost uniformly an angle 3600. However, we can observe only the
galaxies which move
along trajectories inclined to the trajectory of the Earth at an angle smaller
than 900 (assumption 3). In means that we are only able to observe
half of the existing Universe (see “Dark side of the Universe” [4]). Since the Earth rotates around the Sun, its
trajectory changes in relation to the trajectories of the rest of heavenly
bodies. Hence, the boundary of the observed Universe – perpendicular to the
trajectory of the Earth – will change in relation to the actual position of the
Earth on the heliocentric orbit. It means that the most distant galaxies would
appear and disappear in different seasons of the year. Unfortunately, in case
of telescopes positioned on the surface of the Earth, this effect can only take
place for the galaxies which are moving with velocity smaller than the speed of
light by the value 2*10-8c.
Conclusions
The
model presented in this paper describes a much simpler reality than the
hitherto theories. The complex picture of the reality which we observe is no
more the property of the reality itself, but it is a result of the manner in
which we observe this reality. The simplification of the model of reality at
the cost of complicating the observation has already been applied during the progress of science. The best known
example is the transition from the geocentric theory to the heliocentric one,
where the complicating of the observation process – by taking the assumption
that the observation is performed from the moving frame – allowed to
considerably simplify the picture of the Universe.
Accepting the model proposed in this
paper considerably changes our idea of the reality. Time and space aren’t now
the dimensions of the reality assigned in advance, but only certain directions
in FER, which vary with the choice of the observer and the observed body. We are therefore able to describe the
reality with the Euclidean model, in which the dimensions are not deformed.
Derivation of the relativistic dependences for
such reality is much simpler than in the hitherto models, for instance the time
dilation (fig.2 and formula 4) or the Lorentz transformation [2]. Moreover, in
the new model the singularities – known from the Relativity Theory - do not
appear. According to the new model, the singularities are not the real physical
limitations. They are only the effect of the manner in which the observation is
performed.
The new FER model opens new ways for
the progress of science by allowing, for instance, for the accelerating of
particles to the speed of light, which can influence the predicted time of the
interstellar travels. Many of the complex problems, as for instance the problem
of the galaxies recession, receive incredibly simple explanations. What is
more, the FER model finally explains the long-discussed problem of the
existence of tachions, which, according to the model presented in this paper,
cannot exist at all [5].
Aside from any discussions about the
new FER model, there is one final test for the correctness of any physical
theory, namely the experiments. The new experimental effects or the astronomic
observations should give an answer to the question whether the observed time
and space are really the dimensions creating the reality – as it was assumed
till now – or whether they are only certain
projections of the “true” dimensions of FER, as I am trying to prove in
the presented paper.
References
[1] W.Nawrot “The structure of time and the wave structure of the matter” accepted for publication in Galilean Electrodynamics
and scheduled for final GED publication in May/June 2007, http://www.astercity.net/~witnaw/eng2001/supertime.html
[2] W.Nawrot “Is The Space-Time Reality Euclidean?” http://www.astercity.net/~witnaw/eng2001/examplelorentz.htm
(feb, 2000)
[3] Milo Wolff „Origin of the Natural Laws in a binary Universe“ http://members.tripod.com/mwolff/PNASLast.html
[4] W.Nawrot “Recession of Galaxies – simple explanation” (apr, 2002)
http://www.astercity.net/~witnaw/eng2001/recession.htm
[5] W.Nawrot “The rule of conservation of
the space-time interval and the signature of
metric tensor. Problem of existence/non-existence of tachions and
three-dimensional time” (feb, 2000)
http://www.astercity.net/~witnaw/eng2001/theruleof.htm