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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 : Light and gravitation 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. 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 v2.12 published 14.4.2014. email: virtanen.pekka1@luukku.com

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2 Contents, part 2 : Observing in absolute space3 The potential of cell-structured space9 Structure of the lattice lines11 Mass of electric field 13 Appearing of a wave function15 The wave function of an elementary particle19 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 field35 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 body47 The basic quantities of absolute space 47 The Einstein's claims against the ether49 More symmetries50 Components of atom in reciprocal space51 The four-dimensional atom model52 The orbital angular momentum of the electrons in an atom54 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 force61 The mechanics of the Farady's law in the ether 62 Biot-Savart's law 64 Entropy and the ether 64 Summary65 Sources 67

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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.

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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 c² R², 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 v² is the difference between observer's absolute speed c² and the absolute speed w² 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 (= c² ) 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

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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 v² 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.

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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 c² 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'

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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 c² 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 c² - v² 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 c²c² w²w² In this space 4.D is not metric, but also not the direction of the 3D-surface.

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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 = cv + 0 = v A H v c We can see that the relative speed v exists physically as perpendicular component in relation to rest frame.

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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. FiFi 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. FoFo 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.).

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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 Ac² 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 e² 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? AA c² 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<

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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 LcLc 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

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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. vc 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.

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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 = mc² 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 ke² = 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+

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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

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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 c² 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 degrees90 degrees 0 c²

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16 0 c² 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. FF' a b f f² = a² - b², when a b and PF + PF' = 2a. Correspondingly it is valid for the speeds v² = 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 = √ Għ / 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 c² diagonals of an octahedron

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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.

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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.

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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

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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 toto w

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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

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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

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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. m² / m1² = h ² / c ², where h = w, and w is the absolute speed of a particle. v c c w c ² v²v² w ² m1 ² m c y= -m² x² P A B y1= -m1² x ² C h ²

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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

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25 vc 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 c² = v² + w², 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 c² = v² + w². 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 45º 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 c² = v² + w² the vector c corresponding to the quantity c is always parallel to 4.D. c lattice line

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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-metric, 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

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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 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

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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 metric describes the space, which is inclined in an acceleration field outside the contraction area. With help of the metric 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 ) c² - v e ² c² 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 - e² 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 c²vn²vn²

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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 Average of surface inside a body V = v e ² ve²ve² 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

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30 c² 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. veve 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 e² + 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.)

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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

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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 c² - 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 vtvt vrvr 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.

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33 With help of the cell transitions we have got for the time passing of a body in part 1 of DTheory T² = (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 T² / 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.

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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

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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 = v² / c². For the contraction of space in an acceleration field appears from an invariance equation ² = o ² (1 - e² ) = o ² 1 - 2V(r) = o ² (1 - 2MG / r c² ) c²

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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.

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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 / x² 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 / x² 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

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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

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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 perpendicular vectors. In three-dimensional space the mass of a body m(x,y,z) is three- dimensional. When the mass is isotropic, we can reduce the mass mathematically as 1- dimensional or m=m(x) by setting the observer to the x-axis. Often the mass can be presumed as a constant. We add to the mass a new component m(4.D) parallel to 4.D by squaring the mass: m ² = m(x) ² + m(y) ² + m(z) ² + m(4.D) ². When 4.D is observed as square of the speed of a body or v² and we add the fourth dimension to the 1-dimensional mass m(x), we get m ² (x,v ²). The mass depends then on the relative speed v. Mathematically the mass is now 2-dimensional and quadratic. The acceleration caused by any force inclines and moves the body in direction of 4.D so that the accelerated body has a length and also a motion in direction of 4.D. A body in accelerated motion has a length and direction of motion in direction of 4.D. At the same time the set of 3D-coordinates of a body is rotated in relation to 4.D. The 4D-inclination of a body is proportional to the acceleration. A body has thus turned to 4.D, but we can not observe it. In the same way the body spinning around itself has a length in direction of 4.D. This all means the principle of equivalence or we can not distinguish the acceleration or gravitation. The acceleration field of a body, which does not move, is isotropic. The pull force is described with a vector F(x,y,z), where the coordinates x,y and z are vectors. The magnitude of the force depends on the distance from the mass centre and the mass. 4.D The body, which spins around its centre of mass, in inclined to 4.D and its set of coordinates is rotated. 3D-space x y z m F H

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40 The great law of conservation The observer, who moves in absolute space at absolute speed c ², measures for an absolute momentum of a body with himself the value P = (mc)² (new quantity). For the same body, which moves at different absolute speed w ², the observer measures for an absolute momentum P = (m1 * w) ². In absolute space the absolute momentum of a body is invariant. We get the great law of conservation or the first equation of invariance: m1² w² = m² c² = constant In physics P = m²c² is a square of four-momentum in inertial frame of reference, which is an invariant quantity in all sets of coordinates. At small velocities m = constant, and the previous formula is written in form m² w² = m² (c² - v²) = constant, which gives, when c = constant, m² v² = constant. Let's transfer that from absolute space to three-dimensional space by taking a square root, and we get mv = constant. The speed changed to relative quantity, which has now a direction. Thus the conservation law of momentum in three-dimensional space can be united to the conservation law of absolute momentum at low relative speeds. For the moving bodies there exists only one single law of conservation. Let's write m² c² = constant for a body, which moves at absolute speed c in relation to rest frame of light. We transfer here the mass into three-dimensional space and we get mc² = constant. This is the conservation law of energy in three-dimensional space. The law expresses the total energy of a body and it is needed that the the mass m does not move in direction of 4.D, or the mass is a constant rest mass, and that c² is a constant. The quantity c² is also the potential of a body in direction of 4.D. The great law of conservation is m1² w² = m² c². We can write: m1² ( c ² - ( c ² - w ² )) = m ² c ² The relative speed is v² = c² - w² now m1² ( c² - v² ) = m² c² which gives m1² = m² c². Let's transfer this to 3D-space, when c² - v² the mass of a body at relative speed v is m1 = m 1 - v ²/ c²

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41 We notice that the locality of mass means that the mass depends on the speed of a body. When we have selected the square of speed to a coordinate of the fourth spatial dimension, we can mention: In even space the locality of mass is observed only in direction of the fourth spatial dimension. That is why the adding of the fourth dimension to the mass of a body was necessary to understand the locality of rest mass. It does not, however, mean that a body has the momentum in direction of 4.D. When a body can move in direction of 4.D, its mass needs to be expressed as quadratic m².

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42 Perception of the complex space on the 3D-surface Travelling around the loop means moving from one point to another. In four-dimensional space moving is done from a sphere to another sphere. The four-dimensional space can be described using a 3D-sphere in rest frame of light. The 3D-sphere closes the four-dimensional space. Then the observer travels around the space in relation to that sphere. In starting point at the moment t=0 the radius of the sphere R=0. Radius R means for the observer a distance to the sphere, which closes the space or orbit. The observer travels at speed c in relation to sphere, when the rest frame of light moves as expanding surface of a sphere, which radius is R = -ct. We can think that the starting point lies now at all points of the surface. When a half of the way is travelled, R = -cT/2, where T is the time needed to the whole circuit. Then the starting point is as far as possible in the space and the R-radius sphere represents the half of 3D-space. Let's now turn the eyes forward and we will see that the way to the starting point is R=+cT/2. When the tour is done to the half point, begins the starting point to come closer from the opposite direction as a surface of a sphere. The surface comes closer to the observer from the other side or reverse side of space. The radius R decreases and R=0, when the starting point is reached. R= -cT/2R=+cT/2 t=0 t=T All points P of surface of the sphere can now be determined by coordinates P(x,y,z,w). During the tour only the coordinates x,y and z of P have changed. Because the tour was possible to do in all directions of 3D-space, the coordinates x,y and z of P may have got any values during the tour. The coordinate w has been a constant. We can do the tour again so that the observer spreads into the space white smoke at the time interval 0

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43 We will thus simultaneously see both sides of space overlapping and we can not know on with side a particle stands. In the next picture the space is shown in three different forms: -P +P P Loop spaceTwo different spaces or spheres, space and its antispace United complex space P -P A vector is defined in space. The point P lies at the top of a vector in form P(x,y,z,n ²), where x,y,z and n ² are the coordinates of the point. Because the unit vector i of 4.D is imaginary, we get for P two different points +P and -P or +P(x,y,z,n) and -P(x,y,z,-n). The points +P and -P are real points, which are observed in united complex space as one point P(x,y,z,n ² ). The radius of loop-space is parallel to the unit vector i of 4.D. Thus the value n² determines the position of points +P and -P in direction of 4.D In united complex space the points +P and -P represents a concrete positive and negative matter or two protons, which stand in space and antispace. In united complex space both protons are observed as unsigned matter in form P(x,y,z,n ² ). The protons have no difference (but the indirectly observed direction of their spin ). The absolute space is a "quadratic space", where the basic quantities may be negative or positive. The motion equations in such space must be quadratic as well. All motion equations are able to be derived from three equations of invariance, one for each quadratic basic quantity. i observer

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44 When we look at the space from observer's point of view, all observers see themselves to stand at the centre of a sphere in space. All observers see the rest frame of light to come towards and then fly away as surface of a sphere, of which centre they are standing. In united complex space the "smoke" in rest frame of light is understood as light. The light flying with the rest frame can be placed with a mirror to the towards coming rest frame or a reflection is caused. The overlapping spaces are thus connected physically to each other in direction of 4.D through an four-dimensional atom, as later is told. Expanding space We have not so far considered the expanding of space. The space expands so that the number of cells parallel to 4.D increases, but the number of cells of 3D-surface does not change. Let's consider the case of 1-dimensional loop-space, when the space expands. The space expands in all points at the same speed so that all observers see the expanding in the same way. We have got before for the unit vector parallel to 4.D: R w ² = R c ² w ² / c ², where Rw and Rc are the lengths of unit vectors at orbits w ² and c ². When the bodies travel around the space at their orbits and the rotation angle is , we get R ² / = a, where a = constant in all points. The constant a is so called cosmic multiplier. The constant a expresses the relative speed of expansion or expresses, how fast does the space in direction of 4.D expand in relation to 3D-surface. r 2 When a = 0, the space is stable. When a > 0, the space expands. Let's consider the expanding curve of those points, which the observer will meet when he travels along the circle of the loop at speed c in relation to the rest frame of light. The curve is called "Archimedes' spiral" r ² = a and d (r ² ) / d = a, where a 0 is the cosmic multiplier. The curve represents the sight to the quadratic expanding space, when we presume that the light is a trace in rest frame of light leaved by the bodies and the observer in expanding space at his orbit will collide with that trace.

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45 A In such space the observer gets an impression that he is in the centre of expanding 3D-space. In addition he thinks to stand at topmost orbit or at the top of 4.D and all other moving bodies are lower and their time passes slower. Let's presume that the 1-dimensional unit cells parallel to 4.D are longer that the 3-dimensional unit cells at 3D-surface. The reason for the expanding of space is that the number of the unit cells parallel to 4.D increases, which means that their length will necessarily become shorter in relation to cells at 3D-surface. There exists no extra space outside, where the space could expand, and the expanding is then only relative between the perpendicular directions. What will happen, when the cells parallel to 4.D have the same length as the cells at 3D-surface? Does the 4.D get closed and does a new dimension then appear? The value of the cosmic multiplier or the speed of expanding is not known. According to the disobservation law it is not possible to observe directly, because the direction of the expanding speed is parallel to 4.D. Let's consider at the next two different cases, where the cosmic multiplier gets different values. In case A the cosmic multiplier a > R / and in case B a < R / , where R is the length of the radius of space. The radius R is not known. B C In the picture is shown the two spirals for both of observer's directions at orbital motion in relation to the rest frame of light. The observer meets on his orbits the light trace at the spiral. The trace are leaved by the bodies, which have moved past the points of spiral. The trace moves along the expanding circle and is observed there. The spirals get closer and will end in the centre of space, where the time passing has slowed and then stopped. The centre is a limit for the observing. The spirals have in case A not any common point elsewhere than in point A. In case B the expanding speed or the cosmic multiplier a is less than R / and spirals cut each other in point C. Thus both complex spaces seem to have there a common point. The light trace at the spiral causes such an effect that the points of the curve seem to move faster in relation to each other than in reality happens. The more far the body stands from the observer, the faster it seems to fly away on grounds of the light trace (red shift). The red shift is caused, when the time of the bodies at inner orbits passes slower. The effect does not thus describe the real escape velocity. R R

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46 Mass, space and energy We have already considered that the 3-dimensional bodies have no momentum in direction of 4.D. The bodies have, however, the potential W = c² in direction of 4.D. What makes the body to sink in space? The contraction of space needs energy. The positive mass and the density of space are equivalent. We can now think that the energy E = mc ² of mass is placed totally into the density of 3-dimensional space contracted by the mass of a body. The density means here the mutual ratio of lengths of the unit vectors in the space. When we consider the basic particle as a wave package and then reduce the volume of the package, it is the same thing as we would increase the mass of a particle by contracting the space. According to the absolute conservation law the absolute speed of a particle must then decrease correspondingly and the relative speed increase. We have already mentioned that the mass of a particle increases with relative speed. This works also in the opposite way or increasing the mass of a particle by contracting the space will increase the relative speed of particle.. The great conservation law is written for the microcosmos: The contraction of the length of a particle in direction of its motion decreases correspondingly the absolute speed of particle or w ² / s ² = constant, where s is the length and w is the absolute speed. This is the second equation of invariance. By connecting the first and second equation of invariance we get: m1 ² s1 ² = m ² s ² = constant, or the mass of a body changes with the length in direction of the motion. We get from the second equation of invariance for the length: w ² / s1 ² = c ² / s ², and by substituting v ² = c ² - w ², we get s1 = s 1 - v ² / c ² The absolute space produces some effects, which are easy to observe. Next we get one example.

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47 The geometry of kinetic energy of a body It is told that the mass and the contraction of space are equivalent. We got a formula for locality of mass: m1 = m 1 - v ² / c ² From this formula we get with a binomial expansion approximately: m1 = m + 1/2 m (v / c) ². The latter term expresses the kinetic energy cumulated as a mass. What does this strange formula mean geometrically? Let's take a cube of 3D-space around a body and let it then be contracted in direction of the motion of a body. Let's consider the cube as 3D-surface called A. c 2 v 2 A = m A1 = m1 The rest mass m contracts the 3D-surface A. When a body is accelerated to the relative speed v, the surface A is contracted to surface A1 and m1> m. The volume Vm is a proportion of the total energy of a body and corresponds to the rest mass. The other volume left in cube Vv is equivalent to the kinetic energy of a body. Thus the cube Vm+Vv is equivalent to the total energy E = mc ² of a body. The length d of a body in direction of motion is contracted almost linear at low speeds v. When the speed v has increased near the value c, the volume Vv of kinetic energy is about a half from the total energy. c Vm According to the geometry, when the relative speed v increases, the half of change of the energy changes into kinetic energy. When dE = mv², we get for kinetic energy dEv = 1/2 m v². From the picture we get for a volume of kinetic energy Vv = 1/2 mc² * (v / c) ² and then E = Vm + Vv = mc² + 1/2 mv², where m is a rest mass at speed v = 0 and the locality of mass is not considered, when v is small. We can now mention that in absolute space the kinetic energy and the inertia are created by the geometry of space. At higher speeds the 3D-space begins obviously be contracted also perpendicular to the direction of motion and the increase of kinetic energy is not any more linear in relation to the square of relative speed. Vv 4.D motion d

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48 The basic quantities of absolute space We can select, for example, N cycles of a certain electromagnetic oscillator to a measurement unit of time [s]. Then the speed of light c can be expressed in length d, which the light has travelled in one second or during N cycles. According to the D-theory the light is a "trace" in absolute rest frame, through which the observer moves in absolute motion. Then the N cycles form in a space the length S. The time is now expressed in distance S. The absolute speed is then expressed with itself so that the length d, which the light has travelled in one second, is divided by length S, which represents the time. The result is a ration d/S = 1. According to the disobservation law the position of a body in direction of 4.D is not possible to observe, so the speed of light must finally always to express with help of itself and it is a rational number. We have already defined the three equations of invariance, which determine all motion: m² * w² = constant, t² * w² = constant and w² / s² = constant, where w is the absolute speed, which can be replaced with the speed of light c. The basic quantities in these equations, the mass, time and length, are turned out to be local in direction of 4.D or relative in 3D-space. They are relative in the same manner as all motion is relative. According to the disobservation law it is not possible to determine for these any absolute measurement unit and not for the speed of light as well. So in 3D-space we need to determine the standards of measurements for all these three quantities in observer's inertial frame of reference. With help of them and the speed of light it is then possible to express all other quantities. When the speed of light is a rational number 1 at all heights in direction of 4.D, we can express with help of the similar rational numbers also the mass, the time passing and the length of a body. When we substitute into the three equations of invariance the absolute speed w = 1, we get: m² = constant, t² = constant and 1 / s² = constant. When all quantities of physics are able to be derived to quantities mass, time and length, we get logically for all sets of coordinates: Principle of equivalence: The laws of nature must be the same in all sets of coordinates in even motion. We have three equations of invariance and three basic quantities, because we have three closed dimensions in space.

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49 The Einstein's claims against the ether Albert Einstein has in his book "The evolution of Physics" written two claims against the existence of the ether. In place of the ether he proposes the field, which would be a substance. (The substance means here a thing or idea, which does not need to explain with any other thing.) Claim 1. The ether can not exist, because the medium would cause a friction and it is not observed. The space lattice or the ether does not consist of the atoms, but the electrons and positrons. The friction exists between the atoms. The space lattice does not cause any friction. The direction of the lattice is 4.D and a body moves in that direction only in acceleration, when a charged body emits an energy to the lattice. The speed of the lattice is always measured to be the speed of light in all directions. (Michelson-Morley experiment). Claim 2. If the ether does not influence to the motion of matter, there can not exist any interaction between the ether and the particles of matter. Only the particles and the bodies, which includes a component in direction of 4.D, like electrons and positrons, can have an interaction with the ether. The interaction causes an electric and magnetic field and an electromagnetic wave. This all supposes that the particles (quarks) are shared parallel to the dimensions or the main axes as the forces caused by them. The fourth dimension differs from the other ones. It and the lattice causes all electric effects. At the end Einstein writes: " Would there possibly exist interaction between the ether and matter only in optical but not in mechanical events. It would anyhow be quite paradoxical conclusion." D-theory shows that the interaction is optical or electric and in addition that electromagnetic effects are mechanical in their nature. The electromagnetic effects have in space their own direction, which separates them from other mechanical effects referred by Einstein. The arguments like these have seriously prevented to develop idea of the ether. As a result, for example, "Dirac's field" created by Paul Dirac is not taken as a physical fact.

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50 More symmetries According to the disobservation law the radius 4.D of space is not possible to observe. We have considered the issue theoretically and considered that there exists the directions up and down parallel to 4.D. We can not, however, know in what direction are the upside or downside. When we look at the issue from point of view of the positive or negative particle, the issue will change. The sign of a particle determines, which side of the 3D-surface is the upside and downside. When we in addition notice that the coordinates of the lattice lines with the opposite signs, which travel in the opposite directions around the space, are rotated in relation to each other 180º, we get a theoretical idea of the symmetry of an atom. The quadratic space can be "opened" mathematically by taking a square root of it. Then we get a positive and negative space, as in the next chapters is told. 3D-space In the quadratic space the polarization of the basic particles to positive or negative particles can be observed indirectly only on grounds of their spin. The space is divided into the normal space and the reciprocal space like in picture. In the reciprocal space the absolute speeds w are replaced with the speeds 1/w. The observer can theoretical consider the space lattice, which consists of the lines of positive and negative particles. The straight lines are overlapped and stand physically side by side. The space can be "opened" mathematically so that the signs of the particles differs and are found out. The particles are positive or negative depending on if they stand at a layer on antilayer. The 3D-surface forms a symmetry axis for the opened space. The axis divides the space to the normal space and the reciprocal space. The reciprocal space (in the picture) is considered next. In the reciprocal space appears an atom model, in which a part of the particles of an atom travels at a layer, and a part at an antilayer. This symmetric model explains the Pauli's rule for the electrons in an atom. Reciprocal space Normal space 3D-space The reciprocal spaces are opened to positive and negative space 0 c 2 4.D c +4.D

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51 The components of an atom in reciprocal space The previous components, protons, neutrons and electrons build in cell-structured space a whole called an atom. The simplest atom includes one proton and one electron and is called a hydrogen atom. In the atom a proton at 3D-surface binds with help of the lattice lines an electron to itself. The electron moves mostly at the distance of 1 layer from the proton or the nucleus in direction of 4.D. It does not travel around the nucleus, but moves with it in direction of the 3D-surface. The electron lies at the hollow of potential and the projection of the electron is observed around the nucleus. Let's consider next the structure of an atom on grounds of the space model and take a closer look to the spin of an electron. The next picture shows an opened reciprocal space so that only the positive and negative spaces are visible. There is a diagram about the positive and negative half of an atom. The halves are symmetric in 3D-space. The nucleus of the atom includes 4 protons, 4 neutrons, (the neutrons are not shown in the picture,) and in the lattice outside the 3D-space travels 4 electrons. In the picture the positive particles are green and the negative are red. The sign of a particle is determined by the layer. On a layer the sign is positive and on an antilayer it is negative. The sets of 3D-coordinates of the halves are rotated 180º in relation to each other. The lattice lines travel past the nucleus to the right and left. The electrons e- (red) in the picture move with the nucleus at their own layers. The 4.D-components of the nucleus and the electrons e- move in the lattice in four phases. In the picture the negative electron above the positive proton feels the aim of lattice to combine opposite particles and a pull force to the 3D-surface. On the other hand in direction of the 3D-surface the electric field of the nucleus holds the electron e- beside the nucleus p+. e- e+ Layer Reciprocal space e- e+ Antilayer Reciprocal space The halves of an atom at a layer and antilayer are symmetric. The spins of the particles are opposite. Also the negative electrons e- polarize the lattice and the atom is electrically neutral. We have defined a proton of the nucleus to electric positive and an electron at a layer to electric negative. An atom is thus electrically polarized.

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52 When the stack of the electrons of a half of an atom is observed from down in direction of 4.D and the layers are thought to be planes parallel to 3D-surface, we can share the electrons at the different planes as circles around the vertical 4D-axis like in picture. Every layer corresponds to one plane or the surface parallel to 3D-surface. The planes lies outside of 3D- space. The nucleus stands on the 3D-surface and for every electron there exists one proton in the nucleus. In the half of the nucleus the protons stand in the same way at different layers of 3D- space. The layers are in 3D-space spheres inside each other. On the layers the numbers of protons increases with the radius of sphere 1,4,9,16,25... The negative half of the atom stands in four-dimensional space on the surfaces of the spheres, which are turned inside out and have the same centre. It can not be distinguished from the positive half in any way but from the signs of the spins of the particles. In this model the mechanical structure of the auxiliary quantum number is found out. The auxiliary quantum number (or orbital angular-momentum quantum number) describes the distance of an electron from the vertical 4D-axis of the atom. The distance is parallel to 3D-surface. The auxiliary quantum number 0,1,2 are named with letters s,p,d,f. The main quantum number ( or principal quantum number) describes the distance in direction of 4.D. The profile of an atom in opened 4-dimensional reciprocal space The four-dimensional atom model Let's consider in four-dimensional space an atom, which has its layers full of electrons. Let's consider first the positive half of the atom in opened reciprocal space. The picture presents the full electron layers of a half of an atom, which is much more heavy than the hydrogen atom. All the spins of the electrons in the upper half of the atom have the same signs. 3D-surface N M L K 4.D +4.D n = 1 +4.D fdpsdpspssfdpsdpspss

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53 We have already defined the three quantum numbers of the electrons in an atom. They are spin, main quantum number and orbital quantum number. They all define the location of the electron in a four-dimensional atom. The sign of the spin determines the half of an atom, the main quantum number defines the distance of the electron from the 3D-surface and the orbital quantum number defines the distance of the electron from vertical axis of an atom and also the radius of the circle, on which the electrons stand around the vertical axis. The circle is cell- structured and consists of 1-dimensional cells. When all these are determined, there is still left the location of the electron at the circle, which surrounds the vertical axis of an atom. The locations are equal to each other around the vertical axis, but they all have their own direction in 3D-space in relation to the centre of an atom. When one direction z of 3D-space is selected around the nucleus of an atom, the locations are not any more equal in relation to that direction. In addition the directions are quantized. The location of an electron in relation to the direction z is described with the magnetic quantum number m. The number of the electrons at the circle is 2l +1, where l is the orbital quantum number, which defines the circle. The magnetic quantum number m gets as many values as there stands electrons at the circle. It gets the values m = 0, ±1, ±2, ±3,... until to the value ± l. According to the Pauli's rule the quantum numbers of two electron in an atom can not be equal. When the quantum numbers determine the location of the electron in a four-dimensional atom, the rule actually expresses that two electrons can not be in the same place in the same time in absolute space. The location of an electron means here the real place and not the place of the projection of the electron at 3D-surface. Also the concept "state" of quantum physics can be understood as the location of a particle in four-dimensional cell-structured space. A particle transfers from one state to another or from one cell of the cell-structured space to another cell. z m = 0 m = 1 m = 2 m = -2 m = -1 When N = 3 and l = 2, there exists 2l + 1 = 5 electrons at the circle around the vertical axis in an atom. Their magnetic quantum numbers get the values m = 0, ±1, ±2 or together 5 different values. The quantum number m defines the locations at the circle, when the direction z is selected at the 3D-surface. We can thus mention that all quantum numbers of an atom can be described geometrically in a four-dimensional atom model. Each quantum number is uniquely connected with the location of an electron. Vertical axis of an atom 3D-surface

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54 The orbital angular momentum of the electrons in an atom The atom has a vertical axis parallel to 4.D. The axis is two unit cells thick. The vertical axis stands in 3D-space in centre of an spherical atom. The electrons of an atom have gathered symmetric around the vertical axis. The angular momentum of an atom depends on gathering of its electrons. Each electron has its own orbital angular momentum and the angular momentum of an atom is their sum. Not any angular momentum exists at the vertical axis in the centre of an atom. All locations of the electrons, where an auxiliary quantum number has the value ℓ = 0, have the value L = 0 of orbital angular momentum. Vertical axis In cell-structured quadratic space the length r is written (D-theory, part l ) r² = ds or r = √ ds, where d and s are integers, which expresses the length and d = s +1. The orbital angular momentum around the vertical axis is calculated by multiplying the distance r and the quantum interaction ħ. The distance of an electron from the vertical axis is expressed as number of the layers. When in an atom the auxiliary quantum number ℓ means the distance from the vertical axis, we get for the orbital angular momentum The main and auxiliary quantum numbers at the vertical axis of an atom. The direction of orbital angular momentum L is the direction of lattice lines and the dimension is [Js] L = r ħ = √ ds ħ = √ ℓ (ℓ +1) ħ, where ℓ is an auxiliary quantum number ℓ = 0,1,2,3...(n-1). The value comes near to the orbital angular momentum of de Broglie's model L = mvr = n ħ, when ℓ is big. Here ħ = h / 2 . L s p d ℓ = 0 ℓ =1 ℓ =2 n = 3 n = 2 n = 1 rdsrds nucleus L r

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55 The electrons in an atom move towards the vertical axis and back the faster the more far from the axis their position is. The motion makes the electrons in the direction of motion asymmetric and their projections at the 3D-surface are asymmetric too. z L z = 0 L z = ħ L z = 2 ħ L z = -2 ħL z = - ħ The magnetic quantum number m determines the quantization of the direction z of orbital angular momentum. We get for the component L z of the orbital angular momentum L z = m ħ, where m = ℓ, ℓ -1, ℓ -2,... 0,... - ℓ The vector, which describes the orbital angular momentum of the electron in an atom, is quantized in relation to the direction z. In picture N = 3 and ℓ = 2. The calculated projection of an electron on the 3D-surface, when ℓ = 4 and m = 1.

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56 The potentials of the space lattice The internal force F i of the lattice aims to keep the lattice homogenous in all directions of space. In the homogenous lattice F i is constant and the internal potential of the lattice is zero. When, for example, an additional number of electrically positive particles or a positive charge is brought into the lattice, a potential appears. The potential is divided into two components: The vertical potential is parallel to 4.D and causes the magnetic field. The horizontal potential is parallel to all isotropic directions of 3D-space and it causes the electric field. The electromagnetic field means a field, which is a sum of the fields caused by the vertical and horizontal potentials. The vertical and horizontal potentials of a charge are damped along the distance in 3D-space in similar ways. The horizontal potential does not cause any current in the lattice or any motion of lattice the particles parallel to the potential. The magnitude of a horizontal potential V depends on a charge. V = - k q / r, where q is a charge and r is the distance from the charge in 3D-space. The scaling factor k is a constant. Let's define the electric field E so that the electric field is the first derivative of the horizontal potential in the lattice at the distance r in all 3D-directions of a charge q. E = q / ( 4 r ² ), where = 8.85 10 (exp -12) [F/m] is a permittivity. The charges are thus the sources of the potential and of the electric field (Maxwell's 1. law). The potential in the lattice are described in part 2. of D-theory. The horizontal potential in the electric field E around the charge q describes the electric flux through a closed surface S: V 4.D +q r 3D-space = E dS = q / . A force between the charges Q1 and Q2 appears from the aim of the lattice to homogenous. The lattice pulls apart the charges with the same signs and aims to bring together charges with the opposite signs. The magnitude of the force depends on the potentials of the charges and their distance. The force is the first derivative of the potential V in relation to the distance. F = q dV / dx or F = Q1 Q2 / ( 4 r ² ) ( Coulomb's law )

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57 The lattice line can move like in picture in a changing force field as a lattice current and at the same polarize the ether. As a result of the lattice current a lattice charge Qi is accumulated. The lattice charge has a potential V, of which projection at the 3D-surface is a magnetic field. The magnitude of the lattice charge Qi is equal to the charge q at the 3D-surface, which causes the potential. The potential V of Qi is observed totally, when the observer's relative speed to the charge q is the speed of light c. A relative hollow is thus needed to observe the potential V. We get Qi = q v/c, where v is the relative speed and it is proportional to the distance from the 3D-surface. We can understand that at the speed v = c, the lattice charge Qi = q and at the speed v = 0 the Qi = 0. The lattice charge Qi caused by the vertical potential and the lattice current is the reason for the force field M. The force field is the bigger the more far the point of the field stands in direction of 4.D. Let's consider the force between the charges +q and +q, when they both cause a lattice charge Qi. The force has the same form as the force of an electric field. ( Coulomb's law ) : Fm = Qi ² / ( 4 r ² ) = q ² v ² / (c ² 4 r ² ). When we substitute here µ = 1 / c ², we get Fm = µ q ² v ² / 4 r ², where µ is magnetic permeability. We can now define on grounds of the force Fm the force field M for one charge +q so that M is the force field of the lattice charge Qi caused by the charge +q. M = µ q v / 4 r ² and now Fm = qv M Note that the field M is not created directly by the vertical potential, but is created by the lattice charge, which is accumulated from the lattice current. The field M is observed only by an other charge, of which potential causes an similar force field. The field M has a source and it is not curled. We call field B, which is sourceless and curled, the magnetic field. These two properties makes it a "pseudo field". The field B is a "field", to which direction all magnetic dipoles are turned in the magnetic field. The magnetic field B is thus a projection of the force field M on the 3D- surface. The projection must have a central normal. It is the electric current. +q c V The vectors M in picture do not penetrate the 3D-surface, thus on the surface M ·dS = 0 M M ·dS = 0

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58 We get for the field B : B = µ qv u / 4 r ², where u is a unit vector parallel to v x r. The first derivative of charge Q i is a lattice current I d, which is observed as an electromotive force also curled and sourceless in 3D-space. The conclusion: The projections of the force field M and the lattice current I d both parallel to 4.D are in 3D-space curled and sourceless. Amperè's law Let's consider a lattice current on a surface S caused by a charge +q, which moves at speed v. r The vertical potential q/r of the charge +q at the surface dS causes the lattice current I d, which then accumulates to the lattice charge Qi. The surface dS is on a sphere S around the charge +q. The flux through the surface dS is = q dS. Distribution of current J = (r) v(r,t) = q/2 dr 2 S r dt When changes, we get so called displacement current i: i = q/2 dr dS = qv dS = d / dt = I d r dt S 2r S +q v dS We get for a magnetic field B at a circuit of the surface dS, when = 90 degrees (see picture) M = µ i sin / 2 r = µ q v / (4 r ² ). We can write B ·dl = µ d /dt, where B is the density of a magnetic flux. The magnetic field appears from the change of an electric flux d /dt. There is no electric current through the surface dS. In static state a constant number of the charge carriers travels through the surface dS and the electric flux through the surface dS is = 0. The constant current i corresponds to the accumulated lattice charge Qi = q v/c. The lattice charge Qi remains constant in static case and also the magnetic field B remains constant, so we can write for the magnetic field Id V Qi B ·dl = µ i, where i =current (constant). We get as a sum of the displacement current and the constant current B ·dl = µ ( i + d /dt). This is the Amperè's law. We can mention that where the electric flux changes, there appears a lattice current and exists a magnetic field. When d /dt = constant, the magnetic field B remains constant. Between the plates of a condenser, which cuts the conductor, stands a magnetic field caused by the imaginary lattice current I d, although the electric current i = 0. The magnetic field is hold on by the charges q, which come to the plates at speed v and keep on the lattice charge Qi = q v/c.

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59 The direction of the magnetic force Let's consider a charge in set of coordinates 1.D, 2.D and 4.D. The coordinate 3.D is removed from the set. The charge +q travels in the lattice at speed v and creates a potential around itself. The vertical potential creates a lattice charge and a vertical field M with help of a lattice current. The charge -q stands in the picture in place or its speed in relation to +q is u= -v and the distance from +q is described with a place vector r. The field M is parallel to 4.D. Let's define in 3D-space a curled and sourceless pseudovector B = v x r q µ / 4 r ², which is the projection of the field M in 3D-space. The force parallel to the field M means for the charge -q an acceleration, which is defined with help of a pseudovector B. F = -q u x B = q v x B = q v x (v x r q µ / 4 r ² ) 1.D 2.D +q v 4.D (3.D) 2.D 1.D -q +q -q r F M v r (B)(B) F u The distance parallel to the field M means the change of the quadratic absolute speed or the acceleration in 3D-space. The force field M has in 3D-space a perpendicular projection, which defines the direction of the acceleration of a charged body. (The projection of a flux vector and its central normal is already defined in part 1. of D-theory) When the projection of force field M appears from the vectors v and r so that v M and r M, can the projection of M have no other direction in 3D-space than the direction of the pseudovector B or v x r. Then to a charge, which travels in place r to direction v is directed a magnetic force into direction of f = v x B = v x (v x r). According to the disobservation law the direction M can not exist in formula of f, but a projection B is needed. According to the same law the projection B of the field M must be sourceless and curled, because otherwise the direction of 4.D would be observed. The projection of the force field M in 3D-space is the magnetic field B. The vector v stands at the central normal of the projection. According to the definition of the projection the field M must have a central normal in 3D-space or in this case the speed vector v. The projection of the field M is observable in 3D-space only, if the observer moves in relation to the cause of it or moves in direction of 4.D to the area, where the lattice charge exists. v 2

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60 The dynamic properties of the lattice Let's consider the homogenous lattice by causing there a lattice current with help of a positive charge. The charge does not bring more particles to a part of the lattice, but only a force, which polarizes the lattice. The lattice aims to homogenous with help of the lattice current. In the lattice current, which is parallel to 4.D and caused by a positive charge, the particles move towards the charge in the negative lattice line. In the positive lattice line the particles move into the opposite direction. As a cause of the lattice current there appears an internal lattice charge and the lattice is polarized. The lattice current appears always, when the potential of the lattice changes. We have before considered the changing potential in place. When the charge travels perpendicular to the lattice at any absolute speed, appears in a point of the space the change of the potential and the lattice current as well. Let's consider a long charge, which moves in the space. +q The arrows describes the vertical potential V, which affects to the positive lattice lines when the lattice aims to homogenous. In the picture the potential V lies at the distance v (= relative speed) from the charge +q in direction of 4.D. In lower picture are the lattice currents Id and -Id created by the moving charge and the accumulated lattice charge Qi at the distance v from the charge. v Id -Id Into the lattice lines appears an accumulated lattice charge Qi = Id dt = q v/c, where v is the distance of point P from the charge in direction of 4.D and at the same the speed v is the relative speed of P to the charge. The lattice current is observed to zero, when v=0 or the distance is zero. This means that the observer, who moves with the charge, can not observe the lattice current or the lattice charge Qi. When the speed v is the speed of light c, the observer can observe the full value of the lattice charge Qi = q. The forces of an electric field and magnetic field are observed equal, when v = c: -Qi P V Fe = Fm => q ² = q ² v ² µ or v ² = 1 = c ² and Fm = Qi ² = v ² = 1 4 r ² 4 r ² µ Fe q ² c ² The accumulated lattice charge Qi causes a force field or magnetic field. The force field is constant between the heads of the long charge in the picture or it is homogenous, when the charge +q and its potential does not change. When the charge has passed the point P, the lattice charge runs down through the lattice current -Id and the magnetic field disappears. The lattice returns homogenous and there are no traces left from the passing, when the speed v is perpendicular to the lattice or the charge is not accelerated.

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61 The electromotive force The lattice current I d is a first derivative of the lattice charge Qi and its phase is perpendicular to the lattice charge I d = d ( Qi ) / dt. When the lattice charge is described with a flux vector Q i, the lattice current is thus described with a first derivative flux vector I d. Let's consider in 3D-space the projections of these vectors parallel to 4.D. The lattice charge Q i is projected as a curled and sourceless magnetic field B. Its first derivative the lattice current I d is projected as an electromotive force perpendicular to the magnetic field. B (= Q i ) B (= Q i ) emf (= I d ) In picture the density B of magnetic flux changes at the surface S and there appears a curled emf , or the density of the lattice charge in space changes and the lattice current I d appears, which is projected in 3D-space like in picture. In the picture both Q i and I d are the projections B and . The central normal of the projection of Q i is the electric current, which is not shown in picture, and the central normal of the projection of I d is the density B of a magnetic flux. The mechanics of the Farady's law in the ether The Farady's law is written in form: E · dl = - d /dt, where the magnetic flux = B · dS. The change of the magnetic flux creates at a surface dS an electromotive force or the electric field E. = - d /dt. Farady's law describes the projections of the changing lattice charge Q i and the lattice current I d in 3D-space! The lattice current I d can do work as any other current. The vertical potential of a charge causes the lattice current and the lattice current then emits an energy through an electromotive force. When we put a conductive loop in a changing magnetic field, an emf and the electric current are generated in the loop. The electric current makes then a work. When the lattice current I d is accumulated into the lattice charge Q i, must the lattice charge include the work of the lattice current. We can think that the lattice current makes work against the internal force F i. The work is then projected to an energy of an magnetic field, which has the nature of a potential energy. S

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62 As before is shown in connection with the acceleration field, causes the change of the speed or the acceleration for a body an inclination to direction of 4.D. The acceleration is proportional to the angle of inclination. Let's consider next separately the observer's inclination in relation to the charge carriers and the inclination of the charge carriers in relation to the observer. The observer has a surface dS, which stands in magnetic field caused by the direct current i. In the space exists then a lattice charge Qi and its vertical force field M. The surface dS is perpendicular to 4.D, when the surface moves in space at an even speed. Let's give an acceleration to the surface dS, so that he surface inclines in magnetic field to the direction of 4.D. The lattice current I d is observed at the surface dS in connection with the inclination, because the surface dS moves in direction of the lattice charge or the lattice charge Qi = qv/c will change. The lattice current I d is observed only as long as the surface is inclined during acceleration. dSdS + 4.D Qi i When the direction of the acceleration increases the distance dx between the surface dS and the current i, the direction of the inclination is like in picture or the surface moves more far from the current in direction of 4.D. Except the lattice current I d it is also observed that the vertical field M is increased, because the distance h parallel to 4.D is increased. The Farady's law follows from this or the change of magnetic field (projection of M) appears at the surface as curled electric field or electromotive force . In connection of the electromotive force always exists a lattice current I d, because Qi = Id dt = q v/c will change, when the speed v of the surface dS changes. Because the acceleration is absolute, we must consider separately also the change of the speed of the current i. Let the current i be a high frequency ac-current, in which all charge carriers are in accelerated motion. A potential of the lattice charge inclines along the current and the distance h in the picture changes. A lattice current I d is again observed at the surface dS. It appears at the surface as a curled electric field or electromotive force . dSdS + 4.D QiQi i h - 3D-space h M Let the electric field be E = E cos t. The speed of charge carriers is proportional to the E. The density of electric flux is D = E = E cos t, so then D / t = - E sin t. The first derivative describes acceleration of the charges, the number of inclination and the observed electromotive force. -

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63 It is not important, if the surface dS is in accelerated motion or the charge carriers, which cause the magnetic field, are in accelerated motion. In both cases an electromotive force is observed. The inclination of the potential or of the observer creates the base for the electromagnetic effects. The inclination of the charge creates the base for the appearing of electromagnetic wave. The regular inclination of a charge is observed as continuous wave motion. The lattice or the ether does, however, not wave itself, but the inclined charge emits energy to the lattice, which resists the inclination. Because of the structure of the lattice the lattice current is reversible only if it is parallel to the lattice or 4.D. The appearing of a transverse lattice current needs energy and creates a disorder (entropy) and is irreversible. The change stays in the lattice and the lattice travels away at the speed of light. The charge thus emits an electromagnetic radiation as in the next chapter is told.

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64 +q v r B Ø Biot-Savart's law: B = u q v sin Ø 4 r where B is the magnitude of the magnetic field, q is the charge, v is the relative speed of the charge and u is a unit vector parallel to v x r. 2 We can see from the Biot-Savart's law that when the observer comes near the charge in collision direction, the magnetic field is not possible to observe. It is zero, when the relative speed v is parallel to motion of the rest frame from the charge. The magnetic field is observed as greatest, when the observer moves perpendicular to the rest frame coming from the charge. E B Biot-Savart's law We have before mentioned that in the relative motion in relation to the absolute rest frame of light only the speed perpendicular to the rest frame has a physical meaning for the observer. This issue has a concrete meaning, when we consider the magnitude of a magnetic field around a moving charge. Biot-Savart's law describes the magnitude of a field, when the observer travels in different directions. We notice that the magnitude of a field depends on the direction of relative motion. Entropy and the ether The Universe has born one dimension after another and the newest dimension has always included a regular lattice made of positive and negative particles. When a new dimension is created, a new time has started, which has then ended to the closing of that dimension. The disorder or the entropy increases with the time. When the fourth spatial dimension of our Universe later in future will get closed and a new time in 5-dimensional space begins, happens a big change in the entropy of our 3D-space. The space lattice is well organized and its entropy is small. When the new time begins, the ether joins to a part of the 3D-space of today. Then the entropy gets a new chance to increase. The predicted heath death of the Universe is not true! The similar change in amount of entropy has happened last, when the 3.D-dimension get closed and the 4.D-dimension was born. In a loop-space the time has not any start point. The time can be considered to start, when the new dimension is created and to stop, when the new dimension gets closed. When the time ends, a new time begins and there exists one dimension more is the space.

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65 Summary At early times philosopher Plato believed that regular polyhedra have in the world a fundamental meaning. In addition he thought in his gave analogy that we observe only a projection from the reality of the world or shadows on the wall of the gave. Philosopher Baruch de Spinoza, who lived much later, thought that in the world must exist only one substance or the base of reality. These old ideas have found their place in D-theory and are the central subjects of the theory as well. Today the development of mathematics has offered a chance to develop the concept of the physical world. The theory of relativity and the abstract quantum theory are the important new models. Albert Einstein writes in the summary of his book "The evolution of physics". "But what is the medium through which light spreads and what are its mechanical properties? There is no hope of reducing the optical phenomena to mechanical ones before this question is answered. But the difficulties in solving this problem are so great that we have to give it up and thus give up the mechanical view as well." At Einstein's time they tried to place the ether into a three-dimensional spatial space and they believed that the medium transfers the waves mechanically. The light was found out to be a transverse wave and the defining of the light got difficult. They were forced to take rather in use a new substance; the field. Starting points of D-theory are totally different. The space and its cellular structure is the central factor in all phenomena of physics. We can see that the observer’s space appears by coarsening from absolute space and is its emergent property. The probability wave of quantum physics criticized by Einstein is finally replaced with a projection of a particle in nonlinear absolute space and thus the more advanced model of the elementary particles is found out. Determinism returns to physics also in level of quantum effects and coincidence can be forgotten. Also criticism by Einstein against the spooky action-at-a-distance is proved to be well- grounded. The Einstein's Relativity Theory denies the absolute space. However, just the absolute space will produce the Lorentz's transformations. D-theory shows in many ways that the absolute motion and the rest frame are theoretically useable ideas to explain many effects of physics. Thus we can predict that a much simpler and clear model of absolute space created in D-theory will in the future replace the Einstein-Minkowski space-age model. The cell-structured absolute space, which is not unique for the observer, explains several interpretation problems of quantum physics. We have described them before; influence of observer’s consciousness on the measurement process, wave function collapse and non- locality, which is understood with help of quantum correlation. All these problems get their solution through one qeometric space model. The theory, which describes a physical world, must describe also the basics of mathematics, because mathematics is an abstract feature of the world. The space, the basis of the world, is a mathematical concept as well. With help of the space a physical theory gives the meaning and the basics for mathematics.

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66 D-theory is not yet a scientific theory. A reader must see that the model is still unfinished. Many properties of cell-structured space are still open. Anyway it is possible to answer to the question “What is everything?”. The answer is that the answer is impossible to be found. One abstraction always stays left, when the world is observed from the inside. The space model of D-theory seems to work and that leads us to certain conclusions. The conclusions are reader's responsibility. Pekka Virtanen

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67 End of part 2. Sources: W. R. Fuchs:Fysiikka B. K. Ridley: Time, Space and Things Albert Einstein, Leopold Infeld: The evolution of Physics Richard Feynman: QED R.T. Weidner, R.L. Sells:Elementary modern physics H. C. von Baeyer: Maxwell's Demon Jukka Maalampi, Tapani Perko:Lyhyt modernin fysiikan johdatus Raimo Lehti:A. Einstein - Erityisestä ja yleisestä suhteellisuusteoriasta Malcolm E. LinesOn the shoulders of giant P.C.W. Davies, J.R. BrownThe Ghost in the atom Wikipedia An observation: All the dimensions of the world are found in jazz!

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