The structure of time and the wave structure of the
matter
Witold Nawrot
This paper presents the new concept of time –
the SUPERTIME. The SUPERTIME is the time identical for all bodies,
independently of their relative motion. The SUPERTIME, together with the
earlier FER model (Four-dimensional Euclidean Reality), justifies in a simple way the wave
structure of matter and allows to introduce the new way of finding functions
which describe the particles as
waves. The new approach also greatly extends the class of these functions.
Introduction
The nature of time is a problem which lies on the boundary between
physics and philosophy and it still remains enigmatic. On the one hand, time is
perceived quite differently from space; on the other hand, if we use it in formulas, it behaves
similarly to the space dimensions, which suggests that we are living in a
four-dimensional reality. If we treat the time directly as the fourth
dimension, analogous to the three space dimensions, we obtain a model of
reality which is very complicated mathematically and conceptually. In order to
simplify the description of reality presented in the previous paper [1], I
proposed the model of Four-dimensional Euclidean Reality (FER), made of four
identical dimensions. Neither of those new dimensions was a time- or a space
dimension. We interpret three of the directions in FER as the space dimensions
and the fourth as the time-dimension, but those dimensions are not assigned in
advance (as the time or space) but depend on the choice of the pair: observer and the
observed body. The FER model led to a significant simplification of the
reality’s description and to eliminating the singularities, which appear
in the Relativity Theory. On the other hand, the new model required defining
the observation process, which explains why the Euclidean reality without
singularities is observed as the Lorentzian
space-time with singularities. The
singularities taking place in the Relativity Theory are now the result of the
manner of performing of the observation rather than the objective properties of
the reality.
In the proposed model [1], bodies “move” in FER along their
trajectories and during this “motion” they pass certain parts of the trajectories
and only these passed parts
are perceived as time and measured by clocks in the bodies’ frames. If we
use the idea of “motion”, then we should introduce an additional
“time-like” dimension, in relation to which the
“motion” will be determined. This additional dimension will be
called “THE SUPERTIME”. It is common for all bodies. Introducing
THE SUPERTIME will allow us to compare “velocities” of the motion
of bodies along their trajectories, i.e. the speeds of the time flow in frames
of these bodies.
The time and the SUPRETIME
What properties would THE SUPERTIME have?
1.
The main property of THE SUPERTIME would be the ordering of events along
the trajectory in FER – the subsequent positions on the trajectory should
correspond to the subsequent values of THE SUPERTIME. Because quantum move*
along their trajectories similarly to the bodies*, the subsequent
positions of quantum along their trajectories should also correspond to the
subsequent values of THE SUPERTIME.
2.
THE SUPERTIME should flow identically (i.e. with the same speed) for all
quantum and bodies, independently of their relative motion.
3.
THE SUPERTIME must include the fact that the time in the bodies’
frame depends on their relative motion and the time in the quantum frame must
be equal to zero.
The mentioned above properties of THE SUPERTIME are fulfilled with the T-value determined with
the formula:
(1) 
where:
xi – i=1,2,3
space coordinates of the observer, x,y,z
t’
– proper time of the moving body
It means that THE SUPERTIME is not an additional dimension in the sense
of the well-known space-time dimensions. It is a value composed of the space
and the time dimensions. Such a
definition of THE SUPERTIME changes the understanding of the time flow.
Up to now, a variability of events was related only to
the changing of the body’s position along the time dimension. Now the
variability is determined by the change of its position in FER – along
the time dimension (the proper time of the observed body) and along the space
dimensions.
For instance:
For the observer’s frame, where 
, we have:
(2) dT2 =dt’2 =dt2
i.e.
the time measured by the observer is equal to THE SUPERTIME, and the events of
the observer are ordered only in relation to its position on its axis of time
(trajectory).
For the observed body, moving with a certain velocity, THE SUPERTIME is
described with the formula (1). It means that the events of the observed body
are ordered in relation to its position along the trajectory of the body -
t’ - and along the space positions in the observer’s frame- xi.
For the frame bound
with quantum* or with a hypothetical non-zero-mass
body moving with the speed of light (it corresponds to the body
moving in FER along the trajectory perpendicular to the trajectory of the
observer) we have dt’=0;
then
(3) 
In this case, the events of quantum* or of the body are
ordered along its position on the direction (axis) which is perceived by the
observer as the space dimension. We should notice that with the help of THE
SUPERTIME we are able to describe the non-zero mass body moving with the speed
of light, while in the Lorentzian space-time it was
impossible due to singularities taking place for this speed.
For all the three cases mentioned above, the flow of THE SUPERTIME
– dT – was
identical, but the proper times of those bodies were different.
We have to remember that, while in the Lorentzian
space-time the notion of the time- and the space-dimensions were separate
ideas, then in FER the notion of SUPERTIME is identical with the notion of distance passed by the
body (in FER). Each body during the SUPERTIME dT passes the distance dS,
and, using coordinates of the observed space- and time dimensions, it can be
written as follows:
(4) 
Therefore, if we try to define the velocity – here the
SUPERVELOCITY – analogously
to the Lorentzian space-time, we can see that the SUPERVELOCITY of all
particles in FER is the same and equals to V=dT/dS=1
In FER all trajectories are allowed, so there can also exist bodies moving along trajectories aligned to the trajectory of the observer at an
angle bigger than 900 and smaller than 2700. From the
observer’s point of view, the time in frames of the bodies should flow
“backwards”. The situation described above is shown in the fig.
1.
Fig. 1 The trajectory of the
observer and trajectories of other bodies. The trajectory
perpendicular to the trajectory of the observer corresponds to a body moving
with the speed of light. The trajectories inclined at angles >900
and <2700 correspond to the bodies moving backwards in time in
relation to the observer.
THE SUPERTIME which would include all these trajectories can be written
then in a complex form, and
assuming that a time dimension is an imaginary one, it will take the
following form:
(5) 
Since r0 and t’0 are constant values, then,
to make things simpler, we can conduct further considerations in
the frame in which r0=t’0=0
(with accuracy to a constant value), and then:
(6) 
where 
and,
because values r and t’ concern the observation of a body in the system
of the particular observer, the formula
(6) can also be written as follows:
(7) 
where t – the time measured in the observer’s frame, equal to t2 = r2 + t’2
Such a definition of the complex SUPERTIME results from the earlier
definition of the j angle,
which denotes the angle between the trajectory of the body and the
observer – i.e. between the time axes of both coordinates systems; sinus
of this angle denotes the relative velocity of the body [1,2].
The situation described above is shown in fig. 2.
Fig.2. The idea of the
complex SUPERTIME. The j angle determines the relative inclination between the
trajectory of the observer and the observed body. |dT|
denotes the increment
of THE SUPERTIME - T.
The imaginary axis in the fig.2. is chosen along the
trajectory of the observed body, while the real axis denotes the observed distance “r”
in space between the body and the observer. Neither of the dimensions is assigned a priori
as the imaginary or the real one.
The flow of THE SUPERTIME consists only of the growth of |T| - marked in
the figure 2 with the increment |dT| - while it does
not depend on the j angle between the trajectories. It means that the flow of THE SUPERTIME is an absolute quantity**.
Since the flow of time in the frame of the body
– dt’ – and the change of the
distance – dr – treated separately are
relative quantities and depend on the choice of the frame, then their composition, equal to |dT|2=dt’2+dr2
, is an absolute quantity and does not depend on the choice of the frame.
Hence, THE SUPERTIME can characterize the body itself,
independently from any observer.
The complex
SUPERTIME “T”,
can be treated in FER both as the SUPERTIME and as the distance. If we
treat the SUPERTIME as the distance, we can define the complex SUPREVELOCITY,
which can be equal to:
(8) 
According to the definition of the observed velocity and the time
dilation [1], we can see that the
real part of such defined SUPERVELOCITY is equal to the observed velocity,
while the imaginary part describes the observed
velocity of the body’s motion along its own trajectory, equivalent to the
speed of the time flow in the observed frame.
The Quantum Mechanics and the new theory
The wave properties of matter are known for almost a century, but the
unification of the wave- and corpuscular properties of particles in a less abstract way than it would
result from the quantum mechanics still presents a serious problem. The
representation of the particle as a wave propagating in a medium such as space
needs assuming the constant velocity
of the wave. On the other hand, the
particles-waves are moving
in relation to one another, with various velocities, and in the
frame of each particle the time flows with different speed. The resolving of this problem is not simple,
although possible –as proved in the recent papers [3].
Meanwhile, in the model presented in this paper and in [1,2] all particles are moving in FER with identical
SUPERVELOCITY. Therefore, we have no more obstacles for representing the particle
directly as the wave in FER. We can express such a wave as a function of THE
SUPERTIME in a most general way as follows:
(9) y = f(T).
For instance, the simplest wave representing the particle can be
expressed with the following formula:
(10) y = exp(-Tw) =exp(-rw)exp(-it’w)
where w=m0/ħ (in FER c=1) for the described particle, and r – the
distance from the maximum amplitude of
the wave.
Next, in FER, where the singularities do
not exist, t’ for the real bodies can be expressed as a smooth function t’(t,r), so for the case of
the observation of a specified body by the specified observer and for the
straight trajectories, the following formula should be fulfilled:
(11) t’
= t’dt’/dt’=
½ d(t’2)/dt’ = ½
d(t2 - r2)/dt’ =tdt/dt’ - rdr/dt’
the
rest mass of the particle is equal to m0=ħw so the formula (10) can be written as follows:
(11) y
=exp(-rm0/ħ)exp[-i/ħ(m0dt/dt’t - m0dr/dt’r)]
= exp(-rm0/ħ)exp[-i/ħ(Et - pr)]
This is the wave function, already
well-known from the Quantum Mechanics. Hence, the wave function which was used in the hitherto
reality for describing the particle corresponds to the simple wave in FER.
Therefore, it should be possible to describe all the quantum effects observed
in our reality as a result of interactions of the waves in FER, whereas the
macroscopic motions should correspond to propagation of the waves in FER along
differently inclined trajectories. Additionally, the factor exp(-rm0/ħ)
appears here. This factor causes the decreasing of the wave’s amplitude,
with increasing the distance from the particle, and the effect of the existence of the particle would be felt,
in some ways, even at the very far distances from the particle. If the
particles are disturbing space and this disturbance extends to infinity, then
the natural consequence of this disturbance will be acting on the system of
particles in order to decrease the global disturbance – i.e. the forces
described till now as an effect of the existence of fields. If the disturbance
described with the function f(T) was complicated enough, consisting for
instance of a several stretched and compressed regions of space, then we would
expect that at different distances from the center of the wave, different
mechanisms responsible for interaction between particles would dominate.
The way to the
unified theory of field
From the above consideration results the following scenario of
constructing of the unified theory
of field.
1.
First, it is necessary
to determine action which corresponds to the disturbance of FER
2. Next, we should
examine a set of functions of the SUPERTIME – f(T)
– describing various kinds of disturbance of FER and, using the principle
of least action, we should determine forces acting on the system of particles.
3. Function f(T), describing the
shape of the space disturbance, should be chosen in a way that ensures the
domination of different mechanisms describing forces acting on the particles
(or waves in FER), at different distances from the center of the wave. Those different mechanisms would be responsible for different kinds of
interactions between particles.
Therefore, we should try to guess the
right shape of the wave which correctly describes the particle in the FER. I
believe that it is possible and that finding such a function is only a matter of time.
If the theory is true, the different kinds of interactions would be only
an effect of the shape of the disturbance of space in FER, which is perceived
in the Lorentzian space-time as the particle with the
wave properties.
Conclusions
The idea presented in this paper, namely that the
particle is a wave of the space and the field is the result of disturbance of space,
produced by a set of particles, and the conclusions of my previous paper [1], namely
that all particles of the Universe are somehow instantly connected to each
other, are not new. Very similar conclusions were presented earlier in [3], so
it could be said that the two papers – the present one and [1] –
only confirm the ideas presented by Milo Wolff [3]. However, the presented
paper is more then
merely a justification of previous theories.
The main advantage of this paper is the assumption that the reality
– FER – is constructed of dimensions different from the observed
ones, and this greatly simplifies
the description of reality. For instance, the function (12), describing a
particle in the Lorentzian space-time,
can be described in FER in a much simpler form (10). Thus, the further
description of particles should start from a function of the SUPERTIME
describing the disturbance propagating through FER, and then, using the process
of observation, it should be transformed into a more complicated function of
the energy and momentum describing the particle in the Lotentzian space-time.
Of course we can try to guess the shape of the function in the
Lorentzian space-time, and it is not difficult for the simplest cases described
with functions similar to (9) – see [3]. However, in case of more
complicated functions f(T), it would probably be impossible. The wave structure
of matter has already been proposed and described in [3], and the wave model of
a particle was also proposed there.
However, it is hard to believe that all particles of the Universe can be
described with only one type of disturbance of the space. We know for instance
from the physics fluids, that the disturbances of the medium can take different
forms. Thus, the idea of the wave
structure of matter [3] should, in my opinion, be extended for a wider class of functions. Introduction of the FER model allows to find
and examine a wide class of functions describing the particles, which should
now be an effect of more or less complicated disturbances of space. I believe
that it could give a final answer to the question of how the matter is
constructed and how it interacts.
References
[1] W. Nawrot “Proposal of simpler
description of SRT” accepted for publication in Galilean Electrodynamics and scheduled for final GED
publication in May/June 2007 http://www.astercity.net/~witnaw/eng2001/Thenewmodelofreality.html
[2] W.Nawrot “Is The Space-Time Reality Euclidean?” http://www.astercity.net/~witnaw (feb, 2000)
[3] Milo Wolff „Origin of the Natural Laws
in a binary Universe“ http://members.tripod.com/mwolff/PNASLast.html