D-Theory - Model of cell-structured space

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16 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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1





Pekka Virtanen





Studies of physics and mathematics





in University of Helsinki, Finland

D
-
Theory
-

Model of cell
-
structured space



Part




㨠䱩L桴 慮a 杲慶楴i瑩tn

Hypothesis:

In large scale the physical space is a background independent, cell
-
structured, three
-
dimensional surface of a four
-
dimensional hyperoctahedron. It is absolute and quadratic in
comparison to the Euclidean observer’s space. Inside and outside the closed surface exists a
cell
-
structured complex space extending to a limited distance from the surface. Manhattan
-
metric is valid in the space.

(The observer’s space is an emergent property of absolute space. It appears from the absolute
space by coarsened observations and it is different for every observer depending on the
observer’s motion. It is the three
-
dimensional surface of Riemann's hypersphere.)

Abstract
:

The background independent

cellular structure of the absolute space were defined. Appearing
of the observer’s space from the absolute space as its emergent property were described. The
Lorentz's transformations were derived from the space model. The rotations of a macroscopic
stick were proved to be length
-
remaining in a cell
-
structured space.

A solution to the measurement problem in quantum mechanics were proposed. A new
interpretation of wave function collapse and of violation of Bell's inequality were proposed. The
uncertainty principle and the phase invariance of a wave function are derived from the space
model.

The structure of the cell
-
structured complex space outside the 3D
-
surface were defined. The
charge, the spin

and the rotations of an elementary particle and the symmetry groups in the
cell
-
structured

complex space were defined. The geometric structure of the fine structure
constant were defined. Time and the momentum of a particle were quantized. Influence of
gravitation were described on appearing of the observer’s space.

The four
-
dimensional atom model and its all quantum numbers and projections on the 3
-
dimensional surface of the hyperoctahedron were defined geometrically. The accurate values
for proton diameter, Rydberg’s constant and the radius of a hydrogen atom were derived.

The geometric structure of quarks and of the three families of particles were defined.

The locality of mass, length and time were introduced in absolute space with help of the
asymmetric wave function.

It was shown that the electromagnetic fields are caused by the effects of the complex space
and that the model is compatible with the Maxwell's equations

This is the version v2.10 published 6.4.2013.

email:
v
irtanen.pekka1@luukku.com

2

Contents, part 2 :


Observing in absolute space



3

The potential of cell
-
structured space



9

Structure of the lattice lines



11

Mass of electric field




13

Appearing of a wave function



15

The wave function of an elementary particle


19

The standing gravitational wave



26

The longitudinal wave of an acceleration field


31

The wave function and the time in acceleration field



32

The eccentricity of a wave function in acceleration field

35

De Broglie's matter wave of an electron




38

The mass and momentum of body




39

The great law of conservation




40

Perception of the complex space on the 3D
-
surface



42

Expanding space





44

Mass, space and energy






46

The geometry of kinetic energy of a body


47

The basic quantities of absolute space




47

The Einstein's claims against the ether



49

More symmetries




50

Components of atom in reciprocal space



51


The four
-
dimensional atom model



52

The orbital angular momentum of the electrons in an atom

54

The potentials of the space lattice



56

Amperè's law




58

The direction of the magnetic force



59

The dynamic properties of the lattice



60

Electromotive force




61

The mechanics of the Farady's law in the ether


62

Biot
-
Savart's law




64

Entropy and the ether




64

Summary





65

Sources





67

3

Observing in absolute space

We observe the space as 3
-
dimensional isotropic room. The fourth dimension is impossible to
observe. The three spatial dimensions 1.D, 2.D and 3.D are perpendicular to each other in even
space. They are closed, no
-
edged and limited. The fourth spatial dimension 4.D is open and
edged.

Because we know that the space is locally curved around a mass, we can assume that the
space can be contracted and expanded and the space itself transmits these forces.

The fourth dimension is not infinite. It has edges and it is limited inside the Universe. The fourth
dimension exists besides every point of 3
-
dimensional space and it does not increase the
volume of 3D
-
space. We can write for the hyperoctahedron, of which 3D
-
surface is euclidean
and flat in large scale:

The law of disobservation

for an euclidean surface
:
The forth dimension is impossible to
observe. As well the location, length or motion of a body in direction of the fourth dimension is
not possible to observe directly.

We get several logical rules from this law and they are described in the next chapters.

The three
-
dimensional observer's space or so called loop
-
space is the closed surface of
Riemann's hypersphere. The radius R is parallel to local 4.D.

All directions in even motion are perpendicular to 4.D.

Bodies at rest travel on their orbits into any direction in 3D
-
space but always perpendicular to the radius R. A body has
absolute speed perpendicular to the radius R and R is
parallel to the absolute coordinate of rest.

R

Note
! The absolute frame of rest can be defined in two different ways. It can be
(1)

in place or
in rest with the cells of the 3D
-
surface or it can be
(2)

in rest with the light. The rest frame of
light moves in relation to the cells or the layers of the space everywhere into the opposite
directions at speed c, which is observed to have the same value in all directions. The
observation is made by reflecting the light and observer's motion or speed does not influence
on result. In the next chapters we consider the rest frame of light as the absolute frame of rest.
It will produce, for example, Lorentz's transformations and a mechanical model for the time.

The absolute one
-
dimensional rest frame of light is parallel to radius R. An even absolute
motion happens perpendicular to the rest frame and is always a central motion.

4

When a body moves without acceleration into any direction in 3D
-
space, it can in principle travel
around the closed space and return back to the start place. Then the body has not moved in
direction of 4.D.

When a force changes the speed of a body, a body moves in direction of the fourth dimension.
The force makes work against the potential W in direction of 4.D. For the potential is valid

W





R
²
, where R is the radius of the space and c is the speed of light. Radius R is not
metric at the euclidean 3D
-
surface.

In 4
-
dimensional space a body has 4 coordinates. They are named x,y,z and w

²
, where w is the
absolute speed
of the body. (The time is not used as a coordinate here.) A force directed to the
body can change these all. The fourth dimension is thus visible in the movement of the body.
When we have thought to increase the speed of a body, we may have actually braked the
absolute speed of a body so that its location in direction of 4.D has sunk.

On the euclidean 3D
-
surface the 4.D can not be metric, but it proves that the speed of a body is
possible to use indirectly to express the location of a body in that direction. The square of
relative speed v
²

is the difference between observer's absolute speed c
²

and the absolute speed
w
²
of a body
.



v
²

= c
²

-

w
²

( Note also! w

²

= (c
-

v)(c + v) )

3D
-
space

The
relative

sinking:

Observer H sees the bodies A and B to fly
so that the relative speed of B is the biggest.

The absolute speed can be observed in this
frame at 4.D
-
axis. The 3D
-
space is here
contracted as 1
-
dimensional.

The relative speed between the observer and a body is v² = c²
-

w² .

The observer's speed in relation to rest frame of ligth is the same as the measured speed of
light and it is always a constant. Because the absolute speed c² of a body is proportional to the
location of a body at 4.D, we get a rule 1. from the law of disobservation:

Rule 1.

It is impossible to observe the absolute speeds of bodies in relation to the absolute
frame of rest. If the relative speeds of bodies are zero (v=0), the squares of the absolute
speeds are the same (= c
²

) for the observer and for the bodies.

If the absolute speeds or the location in direction of 4.D could be observed directly, would the
fourth dimension be observed as well.

Note! The space, where the relative speed v of a body represents one dimension, is called in physics
phase space
. By means of the phase space can all macroscopic effects of physics be described.

4.D

H

A

B

c

2

w

2

5

When two bodies do not move to each other, their absolute speeds are the same.
When a force
flings the bodies out of each other at some relative speed v, the absolute speeds of the bodies
change, but it is not possible to know how for each body.

Later is shown with help of the space
model that this theoretical change of the absolute speeds depends on the directions of absolute
motions.

We can look at the motion of rest frame of light in relation to the observer theoretically with help
of geometry and algebra.

The rest frame moves in three
-
dimensional space through any point of space so that it comes
closer towards the point from all directions at absolute speed and after passing the point it will
escape to all directions.

All points of the 3D
-
space lie in the same situation. For the observer it is not possible to know
the direction of the rest frame behaving like this. There are two possibilities. The direction may
be forward or reverse. If the direction would change, it would not be observed. The rest frame is
not, however, compressed here to zero size in each point of the space. The rest frame moves
only parallel to main axes and the absolute space is quadratic.

We have got for the relative speed of a body v
²

= c
²
-

w
²

. The relative speed v
²

does not
depend on the signs of absolute speeds w or c and so the direction of absolute speed is not
possible to observe.
Geometry and algebra produce the same result
.


6

The law of disobservation includes the
rule 2
.

All observers measure the same value c² for the square of the speed of light
. Otherwise the
observers could define their locations at 4.D and it would be against the disobservation law. The
squares of the absolute speeds or the location at 4.D of the observer or of a body is not
possible to know. For the observer self it is always c
²
and for a moving body it is always

w
²

= c
²

-

v
²

. The relative speed defines the difference.

The relative speed at the 3D
-
surface and the absolute speed in direction of 4.D are
perpendicular to each other.
We can draw for them a curve, which is a c
-
radial circle according
to the formula v
²

+ w
²

= c
²
. Bodies A and B moves in the picture at relative speed v to each
other.

A

B

c

c'

v

v'



Both the bodies lies at their own circle, which corresponds to their absolute speeds. Both ones
measures a value c for their absolute speed of light. The sets of 3D
-
coordinates of the bodies
are rotated with the speeds of the bodies.

The body A moves at absolute speed c and the body B moves at absolute speed c'. The sets of
3D
-
coordinates are rotated in relation to each other with angle

. The body A observes that B
moves at relative speed v, and B observes that A moves at relative speed v'. The speeds v and
v' are observed equal in 3D
-
space and the angle between them is the same

.

The consequence of the rotated sets of 3D
-
coordinates is that the observer B calculates for the
absolute speed of A the value w', which is less than the B's own absolute speed c'. At the same
the A seems to stand at lower location in direction of 4.D. Both observers A and B calculates
from formula w
²

= c
²

-

v
²

the other one to move at lower absolute speed and to stand at lower
location at 4.D than himself.

The rotation of the sets of 3D
-
coordinates is described later with help of the irregularity or of
eccentricity of an elliptic wave function, which is connected to all bodies and elementary
particles.

A: c v

B: c' v'

w

B'

A'

w'

7

The relative speed v causes a relative sinking of the 3D
-
surface. Observer’s 4.D
-
components
have been rotated to each others, but their 3D
-
spaces are still perpendicular to 4.D. Both the
observers, who move in relation to each other, stands physically above each other in direction
of 4.D. This means that they observe each other's time to pass slower, the rest mass to
increase and the length in direction of motion to shorten.

All quantities do, however, not change in this case. Such quantities are the
invariances

of
absolute space:


c
²

x

t
²


m
²

x

c
²


and c
²

/ s
²

The constant c needs to replace by absolute speed w of a body, when there is a body, which
moves at any relative speed. Then for a body is valid an
invariance equation


c
² t² = w² t' ² .


These invariances describes the fundamental features of space and matter and the three basic
quantities (time, mass and length) both in micro
-

and macrocosmos. As quadratic they also
describes the symmetry of the world. These invariances are considered more later.

Rotation of the sets of coordinates causes
every observer to believe himself to stand at outer
edge of the fourth dimension
. The fourth dimension is edged and is possible to define for it the
inner edge and the outer edge (or lower and upper edge)

We get a
rule 3
.
The speed of light is the biggest possible speed

, because the edge of 4.D is
not possible to cross. This rule is equal with the previous rule number 2. It makes all observers
equal to observe their own location in 4.D.

Simplified picture

of relative sinking of 3D
-
surface:

Every body and particle has its 4D
-
component (red
arrow in the picture).

The two observers see that they stand higher in
direction of 4.D. Their own absolute speed is c and
the other’s is w. The relative speed v gets the 4.D
-
component to rotate in relation to 3D
-
surface and to
each other. Both 3D
-
coordinates stand perpendicular
to 4.D in even motion. Both the observers stands at
the same even 3D
-
surface.

We get for the angle of the rotation between the sets of coordinates:


tan o = v/w = v and w / c = 1
-

v² / c²



-



Later is shown that Lorentz's transformations are realized in space model of D
-
theory. The
model of absolute space even insists it.

4.D

c
²

w
²

4.D

4.D

In this space 4.D is not metric, but also not
the direction of the 3D
-
surface.

8

Rule 4.

of disobservation law:

The curvature of the space in direction of 4.D is not possible to observe and the space
always seems to be flat.

All measures of curvature in large scale produce always the flat
space. If the curvature would be observed, also 4.D would then be observed.

The observer's own location in 4.D is always c² and the height of space in direction of 4.D is
proportional to speed range 0


c

²

. The
height
of space means here the area, where it is
possible to make observations indirectly. It is possible to get indirect observations only at the
limited area in direction of 4.D. According to the law of disobservation it is impossible to
define the
height

as metric. The effect, which gives the indirect observations, is limited in one
way or another in range
0


c

²
. Any direct observations in this direction is not possible to do.
This idea occurs later from the details of the space model and the four
-
dimensional atom
model, for example.

The speed of light is the highest speed and it is used in D
-
theory to describe indirectly the
maximum values of several quantities in direction of 4.D.

The rest frame travels at absolute speed c on the 3D
-
surface through the point A. The
observer moves against point A and against the rest frame at relative speed v. He can now
calculate for the speed of rest frame v + c = c, because the speed of light is always the same.
This means that the rest frame comes from point A to observer H always at speed c
regardless of the observer's relative speed v. The relative speed v has no physical meaning in
3D
-
space in relation to rest frame coming against the observer.

A

H

v

c

A

H

v

c

v + c = c

v + 0 = v

A

H

v

c

We can see that the relative speed v exists physically as perpendicular component in relation to
rest frame.

9

The potential of cell
-
structured space

The orbital motion of the lattice line shapes around the space at speed c creates the potential of
the surface W = c². The potential is parallel to 4.D.

In balance the masses parallel to 3D
-
surface are evenly shared over the 3D
-
surface and create
a pull force around the whole closed surface. When the pull force is parallel to the surface and
surrounds the surface, it has a component perpendicular to the surface. That component means
a potential, which is equal but in the opposite direction as the potential W created by the lattice
lines. The potential causes the inertia of all bodies. (Look at the picture at the next page)

However, the bodies with mass are not shared evenly on the 3D
-
surface and they move at the
surface at different speeds. Then in places, where the mass exists locally more, the balance is
changed so that there is less potential in direction of 4.D. There exists a local hollow of potential
or there exists an acceleration field. The change of potential in 3D
-
space is described with help
of
escape velocity
v
e

of the acceleration field

so that the potential is



W = c²
-

v
e
² = w² .

The cell
-
structured space can be described by the unit vectors. The mass has the ability to
contract the space locally. When the mass contracts the local projections of the unit vectors at
3D
-
surface, the surface will be curved proportionately. It means that the surface has sunk
towards the centre of space. The sinking on the other hand means that the surface is locally
inclined in relation to 4.D. The inclined surface gives acceleration for a body, because the height
of the surface is proportional to the absolute speed w² = c²
-

v
e
² ( = W) of a body. The sinking of
the surface is not metric, but is expressed with quadratic escape velocity v
e
².

The contraction of 3D
-
space in an acceleration field means the shortening of the horizontal local
projections. (D
-
theory, part 2.)

Lets consider closer the position of the 3D
-
surface in the space and its potential.

F
i

Pull force of a mass is parallel to the
surface and causes
F
i to the centre.

Orbital motion of the lattice line shapes at the
speed c causes
F
o out from the centre.

F
o

F
i

= F
o

balance

c

The balance between the forces F
i

and F
o

and its dependence on the speed of light will cause
the changes in magnitude of gravitational field to propagate in space at the speed of light. We
can then talk about the gravitational waves (D
-
theory, part 2.).

10

The total energy E = mc² of a body is related to its location in direction of
4.D. The mass m causes the curving of 3D
-
space, which is possible to
describe as an effective curved area

A

at the 3D
-
surface. The energy E
= mc²



A
c² is then a four
-
dimensional volume.

In an acceleration field besides a body the 3D
-
surface has sunk as a
potential hollow V by a number, which is expresses by the escape velocity
v

of the acceleration field or V = W
-

c² =
-
ve². What happens to the total
energy of a body, which has fallen into an acceleration field?


A




m' = m / 1
-

v² / c² .

We get for the kinetic energy, which equals to energy of changed mass,


E
k

= m'c²
-

mc² = mc² 1
-

1




1
-

v² / c²

, which then gives approximately with help of binomial expansion for kinetic energy



E
k

= mv² / 2

, when v<<c.

This is proved later in D
-
theory also geometrically based on the space model.

A body stands in place in an acceleration field. Its mass is m'. The mass has increased because
of the field. The kinetic energy E
k

corresponds to the escape velocity. Then the kinetic energy is

E = (m'
-

m)c² = E
k.

For the mass m' is valid in potential W = c²
-




m' ² (c²
-

v²) = m² c²

( = the invariance equation of mass ) or

E = mc² =


A


11

When all loops around the 3D
-
surface have the same length and the
position of a body at an inner orbit is marked with the absolute
speed w², we get for the linear relation of unit vectors

L
w

= L
c

w / c or
L
w
²

= L
c
² w² / c²

, where L
w

and L
c

are the lengths of the unit vectors at speeds w
and c. When the space is identical everywhere, must also the unit
vectors parallel to 4.D change in the same relation. We get

R
w
²
= R
c
² w² / c² , where R
w

and R
c

are the lengths of the unit
vectors at the orbits w² and c². These equations describe the
relations of the unit vectors in loop
-
space.

Rc

Rw

L
c

c

w

When the space is identical at all orbital speeds, we can consider, what the time passing must
be at orbital speed w. Let's assume that the lengths s of loop circles are equal at all speeds or
orbits.

The speeds of bodies are c² and w² . The times between events are respectively T
c

and T
w.



v = s / t or w² = s² / T
w
² and c² = s² / T
c
². We get


w² T
w
² = c² T
c
² . We can now substitute w² = c²
-

v², where v is a relative speed.


T
w

= T
c



1
-

v² / c²

The result expresses the locality of time passing, when the location depends on the relative
speed of a body.

We already got for the lengths of the unit vectors parallel to the loop
-
space


L
w
²
= L
c
² w² / c² . When we substitute w² = c²
-

v² , we get


L
w =
L
c

1
-

v² / c² .

The length of a body depends on the relative speed of body.

In even space the lattice lines moving to same direction form on layer and
antilayer a square, which is a circle in observer's space. (D
-
theory, part 1.)
Note! When a lattice line on antilayer is shown in the same picture with a
lattice line on layer, it must be inverted. Then the signs of the lattice lines
correspond to each other. (Or when a particle is moved into opposite
space, it is inverted.)

Structure of lattice lines

12



Schrödinger’s wave equation is globally and locally invariant for changing the phase of wave
function

(x). However according to the previous picture the phase of the lattice and also
the
wave function

(x changes locally in acceleration field. The
wave equation starts to work
when a fixing term is added into it. It will change or fix the phase of wave function locally by
an equivalent number.

This so called
Yang’s and Mills’ fixing term
depict then the
acceleration field in all points of space and has the form A(x)

(x). Function A(x) is here the
potential energy function of acceleration field. A similar but oppositely directed change in the
phase of the lattice lines occurs also in electromagnetic field and also there an equal
potential energy function A(x) can be added to wave equation. When A(x) can depict
different force fields, it means so called
gauge freedom
, which is a fundamental concept in
Standard model.

Besides a layer and an antilayer of the 3D
-
surface exist in direction of a main axis always
four lattice lines, which all are 137 layers long. Together they form a circle like in the next
picture. The lattice line is now considered as a part of a circle, which can be rotated on the
circle as function of rotation angle. Rotation leads to appearing of a photon. After appearing
of a photon rotation does not any more happen.

When the 3D
-
surface is inclined in acceleration field, will also the angle of the lattice lines
change in relation to the 3D
-
surface. The square in the previous picture changes to a
rhombus

and the circle changes to an ellipse to describe the asymmetry of the inclined
space. At the same time the
phase of lattice in the field changes locally
.

The asymmetry of an ellipse is described similar as before
or with the speeds. An escape velocity and the longitudinal
wave of the surface appear on an inclined surface in
potential of the field. The focus point of the ellipse is
determined by the inclination of the surface and it is
expressed with help of an escape velocity v² = c²
-

w².
When the inclination increases, the lattice lines change
more parallel to the surface. The limit is the event horizon
of a black hole, where they all are parallel to each other and
parallel to 3D
-
surface.

When the lattice lines on the inclined surface are not any
more perpendicular to each other, an interaction appears
between them. Interaction resists the inclination of the
surface.

v

c

w

3D
-
surface

The speed vector
c

is also on the
inclined surface always parallel
to 4.D and
v

is always parallel to
the 3D
-
surface.

4.D

The frame of the lattice lines is rotated by an angle


as also the 3D
-
frame
of an accelerated body. If the body is electrically charged, it emits energy
to the lattice lines.

13

Mass of electric field

The total energy E of electrons e
-

and e+ lies in their electric field or in the structure of the
lattice. When energy and mass are equivalent, corresponds the mass of the particles e
-

and e+
to their energy or E = mc
² and appears as curvature of
3D
-
surface. The potential energy E
p

of
electric fields is possible to get calculated, when the radius of electron is known. The radius
limits getting any closer to the centre of potential field. The length d = 2.82 fm is used as the
radius. It is called also for the classical radius of electron. According to Coulomb’s law



E
p

=
-
ke² / r .

Let’s calculate E
p

at the distance of one layer from nucleus, or r = d, where d = 2.82 fm.

We get


E
p

=
-
ke² / r =
-
ke² / d.

Let’s set the known expression from Bohr’s atom model to the previous formula



ke
² =
ħ

c / 137

and we get



E
p

=
-
ke² / d =
-

ħ

c / 137d.

The E of mass is got from the known formula
ħ = 137dmc by multiplying it by the speed c or



ħ c = 137d mc² = 137d E

, where
mc² = E. For energy E

is got



E

=
ħ c / 137d .

Finally we get



E
p

=
-

ħ

c / 137d =
-
mc² =

-
E

.

So we can mention that the potential energy at the distance d from the charge e is the same as
the mass energy of electron, E
p

=
-
E

. The energy of electron thus consists of the electric field
in the lattice and the curvature of 3D
-
surface but noting else.

It is already told, how the mass of proton makes the 3D
-
surface to curve in direction of 4.D up
and down. Also the masses of electrons e
-

and e+ make the 3D
-
surface to curve, but not in
direction of 4.D. The electrons do not stand like a proton on the 3D
-
surface and they can not
curve the surface in the same way. Let’s consider next the way, how the 3D
-
surface curves
besides electrons and other electrically charged particles, which have a 4.D
-
component.

3D
-
surface can be curved also in direction of the 3D
-
surface or the surface is then curled. The
momentary direction of curling has a connection to the direction of lattice current.

X

Y

In the picture the electron e
-

makes the X
-
axis to curl clockwise and e+
anticlockwise, when 3D
-
surface curls and contracts. On the surface occurs
a longitudinal wave parallel to X
-
axis or the surface moves in relation to the
lattice. In the next phase the X
-
axis straightens and then curls again into
same direction. Also Y
-

and Z
-
axes are curled in the similar way. The space
becomes asymmetric in the same way as happens around an uncharged
mass or particle. An acceleration field caused by a mass appears.

The mass of an electron does not cause a hollow on the 3D
-
surface as an
uncharged mass does, but the surface is even in direction of 4.D.

X

e
-

e+

14

Electrons e
-

and e+
curl the 3D
-
surface into opposite directions. When
electrons e
-

and e+
stand in space side by side, their masses do thus not cancel each other out. The sum of
longitudinal waves appearing into 3D
-
surface corresponds to mass of two electrons.

When the lattice current caused by an electron in relation to X
-
axis is considered, we can see
that the lattice currents have opposite directions on different sides of the electron.
Correspondingly the directions of space curling are opposite on different sides of the electron.

e
-

X
-
axis

When we derived before the masses of electron and proton, we considered the ringlike
structure of the projection of electron on the 3D
-
surface. The structure has a connection to the
curving of 3D
-
surface by curling. Instead the 3 quarks of proton do not close themselves to a
ring and therefore the mass of proton occurs as a hollow of 3D
-
surface in direction of 4.D as
later is told more.

Although the mass of electron occurs in the different way as the much bigger mass of proton,
they have similar properties. They both include the longitudinal wave of 3D
-
surface and
asymmetry of space, which creates an acceleration field.

The masses of electrons are small in comparison with masses of protons and neutrons. Later
in D
-
theory in part 2 we consider the acceleration field and its structure on 3D
-
surface caused
by mass.

Direction of space curling on X
-
axis

X
-
axis

Y
-
axis

15

Appearance of the wave function

In 1. part of D
-
theory a proton and a neutron were described as an octahedron or an
antioctahedron, which are contracted frequently and which contains a 4.D
-
component parallel to
a lattice line. A waving particle curves the space around and the oscillation creates a mass for it.
The mass gives for a particle a momentum, which remains. The momentum gets the particle to
progress as a wave directly in a cell
-
structured space. Let us consider next the structure of a
particle in four
-
dimensional space.

In the picture a particle lives in four different phases. The 3D
-
surface oscillates in direction of 4.D at the particle.

Oscillation of a particle is based on its 4D
-
component. At the moment, when the 3D
-
surface is
not contracted, the whole energy of a particle is in the perpendicular motion of 4D
-
component
towards the 3D
-
surface. The mass and energy of a particle are predeterminated by the
properties of 3D
-
surface and 4D
-
component. The mass is caused by the amplitude of the wave
and is a constant only in inertial frame of reference (in the picture). The size of a particle is one
cell.

4.D

1.

2.

3.

4.

3D
-
surface

Cell

The cell is contracted by
curving.

Longitudinal
wave of the
surface

4.D

3D
-
surface





When the particle is fully parallel to the even 3D
-
surface, the longitudinal speed of the surface is
the same as the speed c of the lattice lines changing +c


-
c, and as quadratic c². When
shifted 90 degrees, the particle is fully disappeared from the observer's space and it is parallel
to 4.D. The length of the particle is now described indirectly by the quadratic speed of light c².
The speed of light is the highest speed and it is used in D
-
theory to describe indirectly
maximum values of different quantities in direction of 4.D.

We can use the mass m of a particle as a multiplier only to adjust the macroscopic quantities.
We get for the particle the quantity E = mc², which describes the size of a particle in different
phases. We can describe the particle as imaginary function in direction of 4.D:


E = mc²(sin

t + i cos

t)


, where


is angular frequency
.

The particle can now be described as two rotating vectors in complex space. Their times pass
into the opposite directions. (Compare with the speeds of lattice lines w1 and
-
w2).

E = mc²(sin

t + i cos

t)

0 degrees

90 degrees

0 c²

16

0 c²

Change of the speeds
of vectors

Change of the speeds
of vectors

When the rotation speeds of vectors differs or when for the lattice lines is valid


w1





w2

,

the particle is asymmetric. The vectors, however, change their speeds between themselves,
when the 4D
-
component changes its location on a layer to the side of lattice lines travelling into
the opposite direction. It, however, does not change its location over the 3D
-
surface yet. An
asymmetric particle is elliptical.

We get for an ellipse generally:

Ellipse describes the relations of the speed components in a particle.

The vectors are defined separately for each direction of the main axes. Then each pair of
vectors describes one of the three quarks of a particle.

When a particle is parallel to even 3D
-
surface, the 4D
-
component changes its location over the
3D
-
surface.

When a particle is at its extreme position contracted as a point, the particle however has the
radius, which is longer than zero. The reason is that all the diagonals of an octahedron have
turned in the particle through their centre parallel to each other and are standing side by side at
a distance of limited length. The radius of the particle in this position is called for
Planck’s
length
. The Planck’s length is in cell
-
structured space the smallest possible length. The length
corresponds to the contraction or to the curving of Compton’s wave length of a particle


c

=
ħ/mc to the size of Schwarzschild’s radius 2Gm/c².

F

F'

a

b

f

f² = a²
-

b² , when a


b and

PF + PF' = 2a.

Correspondingly it is valid for the speeds

v
² = c²
-

w², when c


w and

w1 + w2 = 2c. Then

a

c and b

w and f

v and PF

w1 and PF'

w2.

P

The Planck’s length is


=

G
ħ / c³ = 1,6 · 10 m or it is extremely
small in comparison with the size of proton. In this way also the
gravitation constant G is connected to the geometry of space.

-
35

2




diagonals of an
octahedron

17

Hollow V

Inclined surface

Inclined surface and its
projection in linear
quadratic space

We understand that a particle is nothing but a four
-
dimensional undamping wave in the space.
The energy of the wave does not fly to the space around, because only an accelerated body
can cause a progressing wave.

In the first part of D
-
theory is already mentioned that the size of cells in the cell
-
structured
space does not change by stretching. The wave motion of the surface contracts the average
size of the local projection of the particle. When the space is on average contracted at the
particle, the 3D
-
surface sinks with the number of potential V and the surface outside the
particle will incline. Also the local projection of inclined surface shortens or the space is
contracted also around the particle. The quadratic length at inclined surface changes linear
with the height of the surface or with the (escape) speed v².

18

The Invariance equations


m²c² = m'²w² and t²c² = t'²w² and c²/s² = w²/s'²

,where w² = c²
-

v², are valid for all bodies and describe a linear space with help of quadratic
speeds. We get from the first and third equation:


m'² s'² = m² s² = constant.

Increasing of mass m or higher amplitude of 3D
-
surface in different directions, also in direction
of 4.D, gets the length s of local projection of a body to shorten and the body sinks in space.

A particle is shortened or it is contracted inside a massive body, when innumerable particles
together contract the space around and the body sinks in direction of 4.D.

Unifying of many bodies increases the mass and makes the hollow to increase more.
Shortening of a macroscopic length in an acceleration field is caused by average contraction,
which happens separately at innumerable particles (reductionism). The oscillation of the system
of several particles is coherent. More about the hollow and the macroscopic acceleration field
later in D
-
theory, part 2.

s

Macroscopic length s is on average
shortened inside a body. The
density of particles determines the
length.

Oscillation of system or of macroscopic body made of numerous particles is coherent, because
the phase of wave function is globally the same everywhere. The phenomenon is called for
gauge principle. The coherent oscillation creates around a macroscopic body a standing
gravitation wave. The wave has longitudinal and transverse component and they create in an
acceleration field the phenomena for three basic quantities, like time dilation.

When a particle moves in relation to the cells of the surface at some absolute speed w, the
particle is absolutely asymmetric. Asymmetry is impossible to observe, because the absolute
speeds are not observable. In addition the observer himself is asymmetric as well because of
his own absolute speed. Relative differences of asymmetry are instead observable with help of
the change of basic quantities. (for example, the length contraction ), D
-
theory , part 1.

19

The wave function of an elementary particle

As before is told a particle is a 3
-
dimensional wave in the cell
-
structured space.

At the moment t1 and t3 a particle with mass pulls the cell
-
structured space so that the space is
contracted towards the centre of particle. The pull force is the basic force of space and it is
proportional to the amplitude or the distance x from the centre of the mass, F =
-
kx. A particle
pulls the space simultaneously from both directions and a cyclic pull is found out. The pull force
spreads through the space as a potential U =
-
G/x to the environment. G is a constant.

A particle can be described as a wave function with the parameters of speed and place:

v = c sin

t (speed
-
part)

x =


cos

t (potential
-
part)

c = speed of light

t = time

k = constant



= wave length



= angular frequency

U = potential

or as a function G(t) =


cos

t + ic sin

t , where i =
-
1 .

The function resembles the shape of
Scrödinger's wave function

for a free particle and
causes the force F(t) =
-
k


cos

t. With help of this wave function we can look at the relative
motion of a particle first in 3D
-
space and then in a gravitational field or in an acceleration
field.

t1 t2 t3 t4

pull force F

F =
-
kx =
-
k


cos

t

v

x

U



v = c sin

t

x =


cos

t

c

c

t1

t2

t3

t4

20

Let's consider the cyclic motion of a particle, when the direction of motion is the same as the
direction of oscillation. When the relative speed between the observer and a longitudinal wave
is v, the wave function is observed as
asymmetric
. The phenomenon has two reasons; The
wave travels in direction of motion longer way than to the opposite direction. In addition the
square of maximum propagation speed of the wave into both the opposite directions is in
observer's set of coordinates always w ² = c ²
-

v ² and in set of coordinates of wave c ².

Asymmetric wave function, when
the relative speed v > 0.

The set of coordinates of a wave is rotated at the angle


in relation to observer's set of
coordinates.

At the moment t
o

the speed of wave is w and then there exists a connection between the time
passing t and the relative speed v. The picture shows that the cycle time of the wave is
increased. The cycle time t of the wave remains in its own set of coordinates, but the cycle
time t1 in observer's set of coordinates is longer or t1 > t and t1 / t = c / w.

Rotated space

We get for the rotation angle



tan


= v/w = v


c ²
-

v ²

and for the time t1 = t c / w = t


1
-

v ² / c ²

Then we get t1 ² w ² = t ² c ² = constant and we can now express:

The quantity c
²

t
²

remains in rotation of the set of coordinates
.

The observer's own wave function can also be asymmetric, but only the differences of
asymmetry are possible to observe. Therefore the asymmetry is always observed as relative.

v

w

c

t

t1

time t

speed

c

0

0



t
o

w

21

The wave function can be shown as a conic section. When the relative speed v = 0, the
section is a c
-
radius

circle. When the relative speed v > 0, the wave function is described with
an ellipse, witch has the distance 2v between the focal points. (At relative speed v = c we get
a parabola as a limit. )

The wave function is now described with a location function:


r

²

( Ø ) = a ( 1
-

e ² ) , where e = v / c and e < 1


1
-

e cos Ø

and Ø is the circulation angle of the wave function. The length of a major axis of an ellipse is 2c
and the length of the shorter axis is 2w. The energy of a particle is divided into components of
motion and potential energy. The potential energy is drawn in picture. It is asymmetric
as well
.
The
eccentricity of an asymmetric particle

is e = v/c.

The particle pulls the space around it. At the same the 3D
-
surface moves frequently as a
longitudinal wave in relation to the lattice. The pull force is direct proportional to the distance of
particle from the other focal point. The pull force then causes the curvature of space or the
potential U (inclination of the 3D
-
surface).

We can define generally for the mass of a wave function:


m =


F(x,t) dx or the mass is the integral of pull force, where

F(x,t) =
-

k
x

cos

t.

The mass m is a positive direct component of F(x,t).

Note!
The mass must be quadratic
always, when the location of mass can change in direction
of 4.D. This is shown later in D
-
theory. Also the coordinate c² parallel to 4.D is quadratic. The
mass m and absolute speed can have a negative value in bidirectional loop
-
space. Instead in
3D
-
space, when c is a constant, both the mass m and speed v are not quadratic. If we write

E = mc ², we assume that c is a constant and m is a constant
as well
. Let's consider next the
mass of a wave function in different conic sections.

Note! When angle Ø = 0 and speed v


c, the
location function of r(Ø) gets to its value the
infinity and also the pull force or the mass
created by wave function is infinite.

w

v

c

v > 0

w

E

c

Ø

22

Ellipse is a conic section and the cone is an asymptotic cone. Absolute space is quadratic in
comparison to observer's space, so the asymptotic cone is replaced with an other quadratic
cone, of which sides are parabolas.

In space model of D
-
theory the absolute basic quantities
length, mass

and
time

are quadratic.
Thus the quadratic cone, which is based on the parabolas, gives the next expression for the
force F (derivation is at the next page) :


(


F ) ² = k / (1
-

e ² ) , where k is a constant of a particle k = m ² and e = v / c.

We get (


F ) ² = m ² , when v = 0.

In
D
-
theory

the sides of a cone are parabolas. The speed of light c is the maximum of absolute
speed of a body and it is at the top of parabola.

y (=4.D)

y

v

y = kv

y =
-
kv

v

y =
-

kv

2

Asymptotic conic section Conic section of quadratic absolute space

c ²

The integral of pull force in quadratic space is equal to quadratic mass m1² of a particle or

(


F ) ² = m1 ² = k / ( 1
-

e ² ) , where e = v / c. This gives m1 ² = k / ( 1
-

v ² / c ² ) and

by substituting v ² = c ²
-

w ² we get m1 ² = k c ² / w ² .

When for the body v = 0 or w = c, we get m ² = k c ² / c ² = k or we can now write



m1 ² w ² = m ² c ²


Correspondingly for the time of a body we get an
invariance equation
:



t1 = t


, when e = v / c.


1
-

e ²



23

The cone is made of two parabola. The height of a cone corresponds to the square of the
speed of light. In such conic section the mass of an elliptical wave function is considered to be
centralized to one of its focal point depending on its direction of motion. The focal point stays at
the center line of a cone at all speeds.

The mass or


F is in a conic section

inversely proportional

to its distance h from bottom of a
cone. (This is not proved separately in D
-
theory.) The conic section at relative speed v = 0 of a
particle is a circle A
-
P at the top of the cone and its rest mass is m.

When the relative speed of a body is v, the wave function changes from a circle to an ellipse
like in the picture. The ellipse has in the picture the profile of a parabola so that the ellipse
and its centre of gravitation lies at the parabola P
-
B (= y1), and the mass m1 is in the centre
of gravitation. When the mass is inversely proportional to the height h, we get for mass of a
wave function an

invariance equation
:


m1 ² = m ² c ² / w ² = m ² c ² / (c ²
-

v ² ) = m ² / (1
-

v ² / c ² ) = m ² / (1
-

e ² )

When the relative speed v increases towards the speed of light, gets the profile of an ellipse
near to the parabola P
-
C as its limit line and its mass m1 gets near to the infinity and
eccentricity

e ( = v/c) gets near to the value 1.

Note! The behaviour of a wave function is comparable with central motion of a planet at its elliptical orbit, which
is described in Kepler's three laws. The increase of eccentricity e increases the mass of a wave function and on
the other hand the energy of a planet. The difference is the used cone. When the speeds gets near to the speed
of light, the Kepler's laws do not work any more. The limit is the escape of a planet from the event horizon of a
black hole at speed of light. There the orbit is the previous parabola.

In the picture the conic section
describes, how does the
eccentricity

e of an ellipse influence on the mass
of a particle.

e = v / c.

m
²

/ m1
²

= h
²

/ c
²
, where h
= w, and w is the absolute
speed of a particle.

v

c

c

w

c
²

v
²

w
²

m1
²

m

c

y=
-
m
²

x
²

P

A

B

y1=
-
m1
²

x
²


C

h
²

24

Let's consider the place of a particle as a function of time. When the speed of a particle is v in
relation to the observer, the motion is like in the picture.

time

Place

c

w

The observer measures the projection of a wave length of a particle at place
-
axis. The wave
length is decreased because of asymmetry and is



1 =


w/c =


1
-

v ² / c ². =


1
-

e

²

. We get an invariance equation




c ² /


² = w ² /

1 ²


We can presume the next equivalencies:

mass


amplitude of a wave or the integral of pull force

time


cycle time of a wave

length


the length of a wave

It is proved that the next quantities are invariable for a body in rotation of the set of coordinates
in four
-
dimensional space:

c ² m ² = constant , c ² t ² = constant and c ² /


² = constant.

Note! Instead of a constant speed c we can use here the absolute speed w of a body, when the
object is a body, which moves in relation to observer. These all can also be written:

c ² m ² = w ² m' ² = constant, and so on.






1

25

v

c





w

When a body moves in relation to the observer at speed v, becomes the wave function of a
body elliptic and its set of coordinates is rotated by the angle


to the observer's set of
coordinates.

The slope of rotating is
v
/
w

and
v



w
. When always

c
²

= v
²

+ w
²
, the speed vector
c

is also in the frame of a
body always perpendicular to the 3D
-
surface of the
observer's frame. So the speed vector
c

shows the direction
of 4.D in both sets of coordinates.

Observer's
3D
-
surface

4.D

We have considered before the rotating of sets of 3D
-
coordinates caused by the relative speed.
In an acceleration field the 3D
-
surface inclines from 4.D and the sets of coordinates of bodies
inclines with the surface. The set of 3D
-
coordinates of a body is rotated from 4.D only in
accelerated motion. The speed vector

c

keeps its direction also in an acceleration field and for
the speed is still valid c
²

= v
²

+ w
².

In Theory of Relativity the time is the forth dimension. When the time appears from the speed
c
of light, which is always a vector parallel to 4.D, the time concept in Theory of Relativity is
understandable.

In a relative motion at even speed the bodies and the observer move at the 3D
-
surface as
asymmetric. In an acceleration field at an inclined 3D
-
surface instead the whole 3D
-
surface with
its bodies moves forwards and backwards in relation to the lattice at speed v = (escape
velocity). The motion is called the longitudinal wave of the 3D
-
surface. Next we consider the
acceleration field or the inclined 3D
-
surface with help of so called gravitational wave.

3D
-
surface
of a body

v

w

4.D

c

In 1. part of D
-
theory is depicted, how the angle of the lattice
lines turns around an electrically charged particle. An ellipse is
drawn in the picture around the lattice lines. Also in this case the
quantity
c

is parallel to 4.d. A half of the major axis of ellipse is
depicted by vector
c
. Without the electric field then angle would
be 45
º and there would be a circle.

The relative speed
v

depicts the escape velocity of electric field
in the point of space. Also for gravitation force on the inclined
3D
-
surface the same rule is valid as soon is proved:

In speed equation c
² = v
² + w² the vector
c

corresponding to the
quantity c is always parallel to 4.D.

c

lattice line

26

L

The standing gravitational wave

In the next picture a body with diameter of L emits gravitational wave to the right an left. The
moving points in the picture depict the single cells of the 3D
-
surface. The cells move in an
inertial set of coordinates of the body along a track like circle or ellipse. The ellipse appears
here because of the observation angle of a circle track. The size of the body changes with
contraction and expanding in relation to the even space. The size L is thus a medium. The
cells of the body move in a gravitational wave in diections of 3D
-
surface and 4.D. If the body
is considered only in 3
-
dimensional space, it would be expanded and contracted in direction
towards the centre of the body in relation to the even Manhattan
-
space.

4.D

Body

Gravitational wave contains transverse and longitudinal component. A wave is a sum of its
components. A body, which is in rest to absolute Manhattan
-
metrics, emits a symmetric
gravitational wave. The both componets of the wave are in phase shift of 180º. In this kind of
wave a point of the absolute space is moving in an ellipse track in a wave.

Wave direction

To drive the animation use PageUp
-

and PageDown
-
keys in SlideShow
-
state (F5).

The motion direction on opposite sides of a body is opposite. Note that the wave of a static gravitation field
does not transfer energy with it. The parts of a wave are bosons of interaction field as for example photons in
electric field. The surface in the picture is 3
-
dimensional.

Body

4.D

27

When the space waves, there can appear a standing wave. Let's presume a body, which
contracts itself cyclical and the space around itself so that both halves of the cycle cause a pull
force into the 3D
-
surface from the opposite directions towards the centre of the body. When the
body pulls the space with its characteristic force, appears a standing wave inside and around
the body. In the wave the space inside the body is contracted in one direction with an amplitude
A. The space is contracted only inside the body, which area is called a

contraction area
. Around
the contraction area the space inclines because the body sinks with the 3D
-
surface. The space
in centre of the body is contracted most and correspondingly it is expanded most after the pull.
At the same its location in the space in direction of 4.D sinks and rises according to the
invariance equations. The wave has an energy in its square of amplitude A², which is described
by the potential V.

In the space of a body appears a sine wave, which
contracts the space from two opposite directions. (See the
picture.) We get as a quadratic effective value of their
amplitudes

A

A / 2

x

v
e
² / 2

v
e
² / 2

When the length parallel to 4.D is according to the space model proportional to the square of
speed v², the potential or the hollow


is expressed as square of speed
as well

or


= V = A² =
v². Outside a body the potential V means an escape velocity v
e
, because from the equation of
potential energy E = mV and kinetic energy we get


½mv
e
² = mV

and v
e
² = 2V or V = v
e
² / 2 = v². if v
e

= c, we get for the potential V = c² / 2.

A

Symmetric standing wave


V = (A /


2 )² = A²/2.

The potential V describes the average sinking of the body.
The quadratic amplitude of a standing wave is


A² sin² kx + v² cos² kx = A², when v = A.

The wave includes a potential part A, which describes the
change of the length, and the speed part v parallel to the
surface.

Let's consider the standing wave first by presuming that
the wave is
symmetric

so that both halves of the cycles
are identical. This leads to the same result as Newton's
mechanics.

Average
body

28

Before is shown that a wave gets asymmetric always, when the speed increases. That happens
also to the standing gravitational wave. The length changes
proportional the same number

in
halves of cycles. Then the halves of the cycles of a transverse wave are not any more of the
same height.

When we have already got used to describe the asymmetry e of a particle with help of the
relative speed v or e² = v² / c², we can now mention that in an acceleration field the
asymmetry of escape velocity v
e
² / c² corresponds to the asymmetry.


Schwarzschild's metrics describes the space, which is inclined in an acceleration field
outside the contraction area. With help of the metrics we get for the escape velocity of the
field


v
e
² = 2GM / r

.

We get the same result from Newton's mechanics, when the mass m is reduced:


½mv
e
² = mV = m GM/r


½v
e
² = GM/r.

Let's consider next an asymmetric standing wave and its effect on the speed of time
passing.

Asymmetric standing
gravitational wave

The asymmetric wave is described in the picture. For the
heights of the halves of cycles or for the speeds is valid


v
n
²

=
v
e
² or v
n
²

=
v
e
² (1
-

v
e
² )


-

v
e
² c²





We observe that the lower half of the cycle of a standing
wave is smaller than the upper half of cycle.


v
n
²

=
1
-

v
e
² = 1
-




v
e
² c²

The quantity 1
-

e² describes the effect of asymmetry on the
basic quantities in an acceleration field as well in relative
motion.

V = v
e
²



v
n
²

29

The internal distribution of mass of a body determines,
how is the potential inside a body.

The transverse wave contains always a longitudinal
wave, which moves in direction of inclined 3D
-
surface
to and fro. The longitudinal wave is zero in centre of a
regular body (x=0), where the first derivative of
potential V or the local acceleration is zero. Outside a
body the maximum speed of an asymmetric
longitudinal wave is the same as the escape velocity
v
e
of the field. The longitudinal wave gets it maximum
value at the surface of a mass, where it affects on the
time passing most.

Note! The escape velocity v
e
² describes the
longitudinal wave only outside the contraction area.

3D
-
surface

3D
-
surface

Average of
surface
inside a
body

V = v
e
²

v
e
²

Longitudinal wave

According to the model of the wave function the slowdown in time passing in relative motion as
well in the acceleration field is proportional to the speed and the asymmetry. We get for the
slowdown in time passing or for the extension of time intervals in an acceleration field outside
the contraction area of a body with help of the escape velocity


t' ² = t² (1
-

v
e
² / c²) = t² / (1
-

2V / c²).

A standing wave of the 3D
-
surface is found to wave in direction of 4.D in an acceleration field
beside the body. The body lies at every moment at this surface. The result is a hollow


= V =
v
e
²/2, because outside the contraction area the surface inclines (, but is not contracted), and its
local projection shortens at the horizontal plane.

x

0

Contraction area

Contraction of
space

30



The potential of a black hole is outside the hole


V
max

= 2GM / r,

where M is mass, G is the gravitational constant and r
is the distance from the centre of the mass. The
potential gets on the event horizon of a black hole the
value

V
max

= c², when we get for the radius r


r = 2GM / c² ,

which is so called Schwarzschild's radius.

v
e

w

c

The speed of light in an acceleration field decreases because of the local increasing of the
cells by the escape velocity or w² = c²
-

v
e
². When the escape velocity v
e

is



v
e

= 2MG / r , we get for slowdown of the speed of light simply


w² = c²
-

v
e
² = c²
-

2MG / r = c² ( 1
-

2MG / r c² ).

In an acceleration field the length is contracted in all directions of 4D
-
space and the space
is asymmetric in direction of acceleration field.

The standing wave changes into a black hole, when the escape velocity of the field is v
e

=
c.

The escape velocity v
e

is at the 3D
-
surface perpendicular to the slowed
speed of light w or v


+ w² = c². (In addition the speed vector
c

is
always perpendicular to the horizontal plane or v
e

is always parallel to
the surface and w is perpendicular to it.)

31

The longitudinal wave of an acceleration field

It is mentioned before that a particle causes into the space a pull force, which changes the
cell
-
structured 3D
-
space contracted and curved. A particle pulls the space in halves of cycles
simultaneously from opposite directions. When the particles pull the space momentary
towards the field, there appears a motion of the surface to and fro. This motion appears as a
longitudinal wave in relation to the lattice lines, which stand outside the 3D
-
surface.

A body creates in space around itself a standing gravitation
wave, where the 3D
-
surface moves to and fro in direction of
the radius of the field. Because the particles at the surface are
a part of the surface, they move with the surface without any
affects of inertial forces. The inertial forces appear only , when
the particles move in relation to the surface. In the centre of
the body, where the acceleration of the field is zero, no
longitudinal wave is found out.

If a body rotates round its centre, the particles of the body are
asymmetric. Then the longitudinal wave is not directed to the
centre of the body but also to the direction of motion. The
rolling body seems to twist the space with it or the space
round the body is twisted with the motion.

The speed of a longitudinal wave outside a body is proportional to the potential of an
acceleration field and it is biggest at the surface of the body. The local speed of 3D
-
surface in relation to the lattice is v
e
², when v
e

is both an escape velocity in the point of
the field and an amplitude of cyclic wave. Thus the speed v
e

must be added to the relative
speeds v
r

of all bodies parallel the radius of the field. The formula to add the velocities is


u = v
r

+ v
e



1 + v
r

v
e

/ c²

32

When a body stands in place in an acceleration field or v
r

= 0, u² = v
e
² or the speed of time
passing is now calculated with help of the escape velocity of the field v
e
. Time passing slows by
the number


t' ² = t² = c² t²



1
-

v
e
² / c² c²
-

v
e
²

When the escape velocity v
e

= 2MG / r , we get for the time slowing in an acceleration field,
when t is the time outside the field

t' ² = c² t² = t²



-

2MG / r 1
-

2MG / r c²

According to the invariance equations the relative speed of a particle needs to increase, when a
particle gets into smaller room or into contracted space. The relative speed of a particle
increases, when it falls down in an acceleration field. The speed of time passing is thus
predetermined by the relative speed and by the longitudinal wave of the 3D
-
surface.

Let's consider a body in place in an acceleration field, which, however, moves in relation to the
lattice with a longitudinal wave v
r

= v
e

sin

t

to and fro in direction of the field. In addition the
same body moves to and fro in some perpendicular direction to the field at the same speed

v
t

= v
e

cos

t.

Then the centre of the body goes around a local circle or orbit and the particle
stays in the field at the same height, because the tangential average speed is v
t
² = v
e
² / 2,
which is the same as the orbital speed of a body at a circle orbit in an acceleration field. If the
tangential average speed of the particle stays but the particle moves only into one direction at
even speed, it stays still at the same height in the field.

Direction
of the
field

v
t

v
r

In picture vertical motion happens only in relation to the lattice. The horizontal motion
happens in relation to the 3D
-
surface.

The wave function and the time in acceleration field

The time has been defined as the continuous series of events, which are transitions of the
lattice lines past the cells at 3D
-
surface. The time was calculated as the geometric average of
transitions in two loop
-
spaces side by side. On the other hand the time is defined to depend on
the eccentricity of a wave function. Let' s focus now the idea of the time of a particle.

33

With help of the cell transitions we have got for the time passing of a body in part 1 of DTheory


T
² = (n
-

k)(n + k) = n ²
-

k ² , when the observer's time passes the number

T
h
² = n². On the
other hand the ratio of quantities k² and n² is the same as the square of the eccentricity of a
particle e ² = v ² / c ² = k ² / n ². The relation between time passing of a particle and observer is
described with the eccentricity of a particle




T
² /

T
h
² = (n ²
-

k ²) / n ² = 1
-

k ² / n ² = 1
-

e ² , where n ²


c ²

When the number of events is decreased and time passing is slower, the length of time or the
interval between events is inverse for the observer or the time t' between events is increased


t' ² = t ² / (1
-

e ²).

A particle is asymmetric also in acceleration field, although it does not move in relation to a
distant observer.

Why does the particle not start to move after dropping immediately at escape velocity? The
features of a particle and also the space causes the slowness or inertia. After dropping the
particle never can reach the escape velocity and the motion
backwards and forwards partly
continues. Let's consider next more detailed the particle in acceleration field.


When the particle falls down in an acceleration field exactly at escape velocity of the field, the
motion backwards and forwards is zero. The asymmetry of space and the speed of an
asymmetric particle corresponds to each other like even space and even motion.

The force of an asymmetric particle to move in an acceleration field causes the conservative
gravitation field around all masses. This effect can be compared with isolating a particle into
closed box, where according to the wave theory reducing the box size still smaller causes the
increase of relative speed of a particle. In acceleration field the particle directs its force
asymmetrically to one wall, but in even space the force is symmetric to the walls.

34

V

An asymmetric wave function

in a linear potential
hollow

V = ax

The wave length and amplitude depend on the
potential.

The body in potential hollow gets an
acceleration towards the bottom of the hollow.

Reference:

Weidner, Sells: Elementary Modern Physics

x



When the time of a particle becomes slower in an acceleration field, must also the length
become shorter. That does the
contraction

of space exactly means. The length shortens. The
asymmetry of a particle also means the increase of mass, when the particle sinks in
acceleration field. The transformations of basic quantities are the same as when the relative
speed changes by a force. The common factor is that the particle becomes asymmetric.

An example of an asymmetric wave function:

When a force is directed to a body outside of an acceleration field in even space, a body
inclines in its own acceleration field or in its own standing gravitational wave like in the next
picture. The body is accelerated and its front part in relation to force stands then always upper
in direction of 4.D and the shorter longitudinal wave affects there less on the time passing than
in back side lower. Then, for example, the clock runs faster in front part of the body and upper in
direction of 4.D.

F

4.D

F

35

The eccentricity of a wave function in acceleration field

In an acceleration field the wave function of a body becomes asymmetric, because the space is
inclined.
The asymmetry means a relative motion

in relation to the acceleration field and the
body starts to move towards the lower potential. The potential of a field is always a function of
the distance r or V(r) =
-
MG / r. The acceleration of a body in a potential field is


dV(r) / dr = MG / r² .

The eccentricity e=v/c of a wave function describes the asymmetry of a particle. An eccentricity
in any point of the acceleration field is got by calculating
the escape velocity v'

in that point


v' = 2MG / r

, where M is the mass in centre of the field and G is the gravitational constant and r is the
distance from the centre of the field. The eccentricity e is now (cf. Schwarzschild's metric)


e = v' / c or e² = 2MG / r c² .

In the gravitational field of the earth the escape velocity from the surface of earth is v' = 12 km/s
and e = 0.00004. At event horizon of a black hole e = c/c = 1 or the wave function is a parabola.

When the escape velocity is wanted to approach to the speed of light c, must the radius of
mass in centre of field be very short. We get for the critical radius R, when e = c / c = 1.


R = 2MG / c²

The asymmetry or the eccentricity is in a point of acceleration field the same for all bodies of
space.
So all bodies get the same acceleration in field and the escape velocity is the same for
all bodies. The escape velocity v' is able to use now to describe the acceleration field and it
helps to define the
absolute hollow



of a field.





= v'
²

/ c
²

= e
²

,

0 <=


<= 1 .


The hollow


expresses the amount of sinking in direction of 4.D.

The escape velocity v' can be replaced by more common relative speed

v and it is found out
that all bodies, which move at relative speed v, have in space a hollow





= v
²

/ c
²

.

For the
contraction

of space in an acceleration field appears from an invariance equation



²

=

o
²

(1
-

e
²

) =

o
²

1
-

2V(r) =

o
² (1
-

2MG / r c² )




c
²

36

We have considered the relative motion and the eccentricity of a wave function. Because the
eccentricity is involved also in an acceleration field, we look next at both issues.

4.D

3D
-
space

4.D

3D
-
space

B: In an acceleration field the set of 3D
-
coordinates of a body is rotated in relation to 4.D. The
absolute hollow


'

appears in an acceleration field.

4.D

3D
-
space

In an acceleration field the eccentricity e' of a wave function becomes from inclination of 3D
-
surface, as already told. Then in cases A and B the wave function has eccentricities e and e'.
We can show that in both cases the eccentricity means the hollow





=

' = v ² / c ² = e ² , where v is relative speed or escape velocity.

The hollow


and eccentricity e of relative speed are always relative quantities, but the
quantities of acceleration filed

' and e' are absolute.

As told before, the inertia is based on the potential parallel to 4.D,
V = c ² or V


R, where R is
the radius of 4
-
dimensional space. The change of potential

V

can then appear from the
change of speed of a body or the change of the position of a body in an acceleration field. The
change of potential

V

is then always proportional to the change of speed or

V




v, where v
is either a
relative speed

or
escape velocity of the field
. In both changes becomes an opposite
force F=ma, which is proportional to the mass of a body so that m = m', when


=

'.

Because
both hollows


and

' are equal


=

' as expressed with help of speeds, it means that
the
inertial mass m and the gravitational mass m' of a body are equal
as well.

The hollows


and

' are significant so that in a hollow the observer's time passes slower and
the mass of a body has increased. The hollow is thus observed indirectly. In cases A and B the
time passing becomes slower by the same number in hollows


and

'.

A: In relative motion the sets of 3D
-
coordinates are
rotated in relation to each other. Still their 3D
-
space is
always perpendicular to 4.D and the observer believes
himself to stand in absolute space above the other
ones. The
relative hollow



is created and it is not
possible to know the real heights of the positions of
bodies in direction of 4.D. It is known that
during

acceleration the position of a body changes absolutely.

The relative hollow in
space.

The absolute hollow in
space.

37

The direct component of the pull force of a body directed to the space damps out with the
distance x or F = k / x. Thus the body sinks in space in direction of 4.D to the height, which
corresponds the contraction of space caused by the body. The curve of sinking follows the
contraction of space and is also in form f(x) =
-
k / x. The hollow of potential is found out.

x

f(x) =
-
k / x

The observer is inclined in relation to 4.D near the massive body, which causes the acceleration
field. The inclination means the acceleration.

The inclination or the acceleration is the first derivative g = GM * 1 / x
²
of the sinking curve
.
This gives the gravitation force Fg directed to a body, which mass is m:


Fg =
-
GMm * 1 / x² , where G is the gravitational constant.

The change of sinking curve is


df/dx = GM * 1 / x
²


and it means at the same the acceleration g.

So that this formula would work also at large distances x, the space should be infinite. It
,however, is not. The space is closed structure and limited in size.

When the space is contracted in acceleration field, it must correspondingly stretch somewhere
else. Because the space is limited, the previous formula can not work completely at large
distances. Because of the stretching of space the gravitation force does not decrease at large
distances so much as the previous formula insists. There exists observations, which support
this idea.

4.D
-
hollow

F

F(x) = k / x

0

M

38

De Broglie's matter wave of an electron

In three
-
dimensional space we observe that when a body moves, its size seems to change.
A body growing apart seems to get smaller. The length of a body seems to be halved, when
the distance doubles.

According the law of disobservation we can not observe directly the fourth dimension as a
distance. However, a body, which has a relative speed v in relation to the observer, is at
certain distance from the observer in direction of 4.D. We do not observe a three
-
dimensional
body to get smaller in direction of 4.D.

If we look at an electron, which moves at relative speed v, we observe the wave length of an
electron to be halved, when its speed is doubled. We can write for the wave length of an
electron according to the de Broglie's hypothesis :




= h / m
e
v ,

where h is Plack's constant, m
e

is a rest mass of an electron and v is relative speed.

According to the D
-
theory the change of a wave length is caused, when the electron grows
away in direction of 4.D so that wave length of an electron looks shorter. The observation is
indirect and we can not calculate the absolute metric distance in direction of 4.D.

The electron and the positron are both particles parallel to 4.D. Their distance in direction of
4.D can be observed as function of relative speed. It is not important, if the speed of a particle
or the speed of observer is changed, because the distance changes in the same way in both
cases. Such particles like a proton and a neutron includes also a component parallel to 4.D.
The projections of these particles are considered more in the third part of D
-
theory.

According to de Broglie's hypothesis the wave length depends also on the mass of a particle.
The mass of a particle causes an absolute hollow in space for a particle, in which case the
distance of a particle from the observer grows and the wave length gets shorter.

More details of the projection of an electron is told in the third part of D
-
theory.

H

39

The mass and momentum of a body

A point mass causes in 3D
-
space an acceleration field and a pull force.

Let's presume that the distance from the mass centre is one unit, when the force is proportional
to the mass only. According to the Pythagoras we can write:

F(x,y,z) <=> m ² = m(x) ² + m(y) ² + m(z) ², where m(x),m(y) and m(z) are the lengths of the