A rigorous definition of new space in the h-space theory

I rewrote the definition of new space of the h-space theory in more rigorous way, see second post from 24.06.2018. Here is a quote of the definition and  its comparison with Euclidean space.

DEFINITION OF SPACE

Volume contains two concepts: space of certain dimension and length. Length and energy are primary concepts. Neither energy, nor length can be reduced to a more elementary concept. For space we will use Hegel’s definition, which characterize space primarily as three dimensions. “Space has, as the concept in general (and more determinate than an indifferent self-externality) its differences within it: (a) in its indifference these are immediately the three dimensions, which are merely diverse and quite indeterminate ”(G .W. F. Hegel. “Philosophy of Nature”, § 198 ). According to Hegel’s ontology, space and time are interdependent. “This disappearance and regeneration of space in time and of time in space is motion; – a becoming, which, however, is itself just as much immediately the identically existing unity of both, or matter ” (G.W.F. Hegel. “Philosophy of Nature”, § 203). Since above, time was reduced to concept of motion that means not time, but motion exists in unity with the space. Before analysis of this unity in details, next we will look more thoroughly to existing definitions of space, or, more exactly, n-dimensional linear space in geometry. The n dimensional linear space is a direct descendant of Euclidean geometry, which is the geometry of real three-dimensional space used in physics.

“Definition 1. A linear space is n-dimensional if it contains a linearly independent system consisting of n vectors, and any system consisting of a large number of vectors is linearly dependent. The number n is called the dimension of linear space. Thus, the dimension of space – it is the largest number of linearly independent vectors. ”(N. V. Efimov and E. R. Rozendorn, “Linear algebra and multidimensional geometry”, 1969, page 35, Russian edition)

“Definition 2. The system of vectors a, b, c, … q is called linearly dependent if there is the equality

αa + βb + γc + … + χq = 0,

where from the numbers α, β, γ, … χ at least one is non-zero.

Definition 3. The system of vectors a, b, c, … q is linearly independent if the equation

αa + βb + γc + … + χq = 0,

is possible only if α = β = γ = … = χ = 0.”

(N. V. Efimov and E. R. Rozendorn, “Linear algebra and multidimensional geometry”, 1969, page 25, Russian edition)

“Zero space. Let L is the set consisting of only one element. What this element is we do not care. We denote it by the letter θ. We define for set L linear operations, assuming that θ in the sum with itself gives the θ and when θ is multiplied by any real number, we also obtain θ. It is easy to see that in this case, the requirements of the axioms I) – 8) are met. Thus, a set L is a real linear space consisting of a single, zero element. Clearly, with the same result set L can be defined as a complex space. Note. All other (real or complex) linear space must have an infinite number of elements. ”(N. V. Efimov and E. R. Rozendorn, “Linear algebra and multidimensional geometry”, 1969, page 16, Russian edition)

Since dimensions originated from the framework of three-dimensional Euclidean geometry, this forces us to consider other dimensions of space in a similar way. In this case, zero-dimensional space, as defined in the cited text, is not realistic, since the vector of zero length does not match anything, i.e. a physical object of zero length does not exist. This means that either any zero-dimensional space does not exist and is just a convenient mathematical construction, or the definition of zero-dimensional space requires revision. A revision is obvious if we look closely at the above quotation. There, the definition of zero-dimensional space does not follow from the general definition of the dimension of space. Zero-dimensional space introduced as a separate definition corresponding to the eight axioms of linear space. Let’s try to define the zero-dimensional space based on a common definition of the dimension where “the dimension of space – it is the largest number of linearly independent vectors”. Three-dimensional space requires three linearly independent vectors, two-dimensional – 2, one-dimensional – 1, then zero-dimensional – 0. Accordingly, zero-dimensional space is characterized by the absence of linearly independent vectors, and that also means that all vectors of zero-dimensional space are linearly dependent. Additionally, the vectors, as real objects of zero-dimensional space, should have non-zero length; the same as all other vectors in spaces of any dimensions.

DEFINITION OF NEW SPACE IN LINEAR ALGEBRA

Definition 1. The length (modulus) of a vector of n-dimensional space of any dimension is not zero.

Generally accepted definition of vector as a directed segment corresponds to the object of non-zero space, for example, the object of three-dimensional space. Such an object can be defined as a linearly independent vector.

Definition 2. Object of non-zero space is a linearly independent vector. A linearly independent vector is a vector of certain length in one direction.

Definition 3. The following 7 from 8 axioms of classical linear space are applied to linearly independent vectors of new space.

  1. Associativity of addition: a+ (b+ c) = (a+ b) + c
  2. Commutativity of addition: a+ b= b+ a
  3. Inverse elements of addition: for every a∈ V, there exists an element a∈ V, called the additive inverse of a, such that a+ (-a) = 0.
  4. Compatibility of scalar multiplication with field multiplication: αa) = (αβ)a
  5. Identity element of scalar multiplication: 1a= a, where 1 denotes the multiplicative identity in F.
  6. Distributivity of scalar multiplication with respect to vector addition:  α(a+ b) = αa+ αb
  7. Distributivity of scalar multiplication with respect to field addition: (α+ β)a= αa+ βa

where a, b, c arbitrary vectors in V, and α and β scalars in F

The mentioned above linear dependence of the vector of zero-dimensional space means that for a given vector, a0, having non-zero modulus, |a0|, the factor α must be different from zero, and the product of the vector, a0, by a factor, α, should be zero.

αa0 = 0

This is possible if the vector, a0, has a non-zero modulus of length, |a0|, and is oriented in such a way that in any given direction vector length is zero. In other words, the vector is located simultaneously in two opposite directions.

Definition 4. Object of zero-dimensional space ais a linearly dependent vector. Linearly dependent vector is a sum of two linearly independent vectors opposite to each other in any direction.

αa0 = α(–a+a)

where,  |a0| = 2|a|

Visually, in three-dimensional space an object of zero-dimensional space, as a linearly dependent vector, corresponds not to a segment but to a ball, with a radius equal to the half of length of the vector, and it is directed away from or toward the center. In two-dimensional space – the circle of that radius, in one-dimensional space – the straight line of the length of the vector. In zero-dimensional space, an object of zero-dimensional space can be represented as a curve of arbitrary direction, whose length is the length of the object.

Following from the definitions above, an object of zero-dimensional space can also be defined as an undirected/ omnidirectional segment.

Definition 5. A linearly dependent vector is a non-directed/omnidirectional segment, collinear and opposite to any vector.

Thus, according to the presented concept of new space,zero-dimensional space is not a single vector/object of zero length, but a set of linearly dependent vectors/objects having non-zero lengths. Since the object of zero-dimensional space is defined as a sum of two linearly independent vectors opposite to each other in any direction for any dimension, zero-dimensional space can be defined as space of any dimension, i.e. it should be omnidimensional,

Definition 6. A zero-dimensional space is omnidimensional space and all n-dimensional spaces are subspaces of the zero-dimensional space.

From this definition, new concept of space implies that zero-dimensional space contains three-dimensional space. I.e. three-dimensional space is subspace of zero-dimensional space. For Euclidean geometry the situation is opposite, three-dimensional linear space includes zero-dimensional space. All features of this new space are presented in the next chapter, “METAPHYSICAL FOUNDATION OF h-SPACE THEORY”.

DEFINITION OF NEW SPACE AS PHYSICAL SPACE

Since the definition of a vector includes, besides the length (modulus), the notion of direction, the concept of a vector in physics is unthinkable without the concept of motion, defining the direction of motion. This allows to redefine the dimension of physical space as the maximum number (n) of independent motions, instead of linearly independent vectors.

Definition 7. The dimension of non-zero space in physics represents the maximum number (n) of independent motions.

This means, for example, that in a three-dimensional space, the motion of any object is a linear combination of its motion in the opposite directions of each of three independent motions, two-dimensional – a combination of two independent motions, and in the case of one-dimensional space – a combination of movements in opposite directions of one motion. In the case of zero-dimensional space, its objects are moving in all directions, i.e. visually it is similar to an expanding balloon.

In a formalized form, the n-dimension can be defined as n‑dimensional volume, i.e. to the power of n – ln.

In classical physics, space is independent from matter. In the general theory of relativity (GTR), space depends on matter, the curvature of space-time is generated by the energy and momentum of matter. In the proposed concept of space, the space-matter dependence is more fundamental, since there is nothing but space composed of objects. Space is discrete matter, where a material unit is an object of space, while time is a relative motion of the objects.

COMPARISON OF EUCLIDEAN SPACE AND NEW SPACE OF h-SPACE THEORY

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EXPERIMENTS TO TEST h‑SPACE THEORY

One more experiment.

ABSORPTION OF PHOTONS BY ELECTRONS AND REFLECTION OF PHOTONS BY POSITRONS

Background. In the proposed theory, electrons absorb photons. This causes an increase in the velocity of the electrons. Positrons reflect photons without changing the velocity of the positrons.

Scheme of experiment 2. The light reflected from the surface of the positively charged metal plate is expected to be more intensive and with less spectrum changes than the light reflected from negatively charged metal plate. The simplest set for the experiment can include:

  1. Laser pointer (red light),
  2. Aluminium plate,
  3. Сapacitor to charge the aluminium plate,
  4. Light intensity detector.

So, the luminosity of the laser light reflected by the charged aluminium plate can be measured by the light intensity detector.
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APPENDIX 2. EXPERIMENTS TO TEST h‑SPACE THEORY

THE GRAVITATION EFFECT ON MAGNETIC FIELD AND MAGNETIC INTERACTION

Background.In the proposed theory, the electric charge is determined by the density of undirected n=0-objects(I), ρ0. And the increase of density ρ(increase of gravity) corresponds to a decrease in ρ0, i.e. a decrease of the elementary charge. This means that magnetic field and, consequently, magnetic interaction depends on the gravitational potential.

Scheme of experiment. The repealing force at the same distance between two permanent magnets should be measured at Earth’s surface and at some distance from it. This force should be stronger at some distance from the surface than it is at the surface.

THE CHARGE-TO-MASS RATIO OF A FREE PROTON

Background. In the proposed theory, both charge and mass of the electron are 459 times larger than that in the textbooks. Because of the electron charge increase, the charge-to-mass ratio of a free proton is 9 times less than that of a free electron. From textbooks this value should be 1/1836. This value was first determined in electrolysis experiments and then in experiments conducted by Willi Wien and J.J. Thomson. According to the proposed theory, in Thomson’s experiments the charge-to-mass ratios of monovalent carbon and oxygen were mistakenly assigned to the charge-to-mass ratios of monovalent atomic hydrogen ion and molecular hydrogen ion.

Scheme of experiment. The Thomson’s experiment should be modified for a chemical detection of the spots corresponding to “hydrogen” ions. For this, the chromium metal plate as a detection screen can be used. If the proposed theory is correct then the spots will have different color corresponding to chromium oxide and chromium carbide. The chemical reaction of hydrogen with chromium should not give the colored spots.

THE IONS CHARGES

Background. In contrast to the textbooks, in the proposed theory the charges of monovalent ions of different chemical elements are different.

Scheme of experiment. Two bodies of symmetrical shape made of aluminium and lead can be charged while they are in a direct contact. This will generate their equal charges, but the different numbers of electrons/positrons should be present on these bodies. Their discharge to the ground will result in flows of different amounts of electrons, i.e. in the electric impulses of different amplitudes or in different stain intensities in the electrolysis experiment with the electrochromic electrolyte.

VERIFICATION OF THE ELECTRON-POSITRON COMPOSITION OF PROTON AND NEUTRON

Background. In the proposed theory, protons and neutrons are made up of electrons and positrons.

Scheme of experiment 1. The existing facilities can be used to conduct experiments on the collision of low-energy electrons and positrons. According to modern physics, these energies would be too low to form protons and antiprotons. However, the proposed theory predicts that protons and anti-protons, as well as other particles, would be formed as different complexes of the positrons and electrons.

Scheme of experiment 2. In the experiment the collision of high-energy electrons is conducted. According to modern physics, collision of high-energy electrons can result in the generation of particle-antiparticle pairs. In the proposed theory, the collision of high-energy electrons cannot result in production of any particles except of electrons.

Scheme of experiment 3. In the experiment positrons emitted by radionuclide (22Na) are directed to a negatively charged metal target. According to the proposed theory, the interaction of positrons and electrons can result in the formation of protons and then hydrogen atoms. The detection system in this experiment should be designed to control the amount of hydrogen atoms.

ABSORPTION OF PHOTONS BY ELECTRONS AND REFLECTION OF PHOTONS BY POSITRONS

Background. In the proposed theory, electrons absorb photons. This causes an increase in the velocity of the electrons. Positrons reflect photons without changing the velocity of the positrons.

Scheme of experiment. The separate traps for electrons and positrons should be used. Then, electrons and positrons are irradiated with photons and changes in the velocities o electrons and positrons should be determined.

VERIFICATION OF COULOMB’S LAW AT DISTANCES BELOW 10METER

Background. The proposed theory suggests that Coulomb’s law is not correct at the distance less than ≈ 105 m. There is a direct proportionality of electrostatic attraction/repulsion to the square of distance, i.e. the velocity of attraction/repulsion between electrons/positrons does not increase (according to modern physics), but it decreases in the range from ≈ 105 m to ≈ 1010 m. If the distance is in the range from ≈ 1010 m to ≈ 1015 m, the velocity of attraction/repulsion of electrons and positrons is zero.

Scheme of experiment. The following experiments can be conducted to validate electrostatic attraction/repulsion at distances less than ≈ 105 m. An electron source is placed behind an electron filter with micrometer holes. The distances between pairs of holes range from millimeters to nanometers. Beyond the filter, a screen is placed which can register spots left by electrons that have passed the filter. The test allows a determination of electron spots with the distance between the holes in the filter. According to the suggested theory, reduction of the distance between adjacent holes down to 10meters will cause increasing repulsion of the electron beams, and the electron spot on the detector screen will increase. But at distances less than 105meters, there will be less repulsion between electron beams, and the resulting spots will be smaller. The maximum spot size must be at a distance of 105meters. In contrast, according to modern physics, the repulsion of electron beams will increase with decreasing distance up to the collision of beams. Accordingly, the spots on the screen will also increase with decreasing distance between the holes in the filter.

Rather than pairs of electron beams, it would also be possible to use separate electron beams but with different diameters (diameters of holes in the filter). According to modern physics, with decreasing beam diameter, dispersion of the electron beam should increase due to mutual repulsion of electrons in the beam. Accordingly, the ratio of the spot diameter on the screen (for the same distance between the filter and the screen) to the initial diameter of the beam on the filter will increase. If the proposed theory is correct, then at beam diameters less than 105meters, the dispersion of electrons will decrease, as will the ratio of the spot diameter on the screen to diameter of the beam on the filter.

THE CHANGE OF CHARGE NEAR NEUTRAL MOVING BODIES

Background. According to the proposed theory, the motion of electrons and positrons changes the density of n=0-objects(I) ρ0. As a result, a magnetic field is generated that consists of n=0‑objects(I) of density ρΔ,and directed according to the sign of the charge. In the case of the motion of a neutral body, there is also a change in the density ρ0, but n=0-objects(I) are not directed. The number n=0-objects(I), displaced by charged or neutral bodies depends on the speed and density of the bodies. Since the density ρ0determines the electron charge, then in the region near moving neutral bodies the value of density ρ0will by changed, i.e. the electrical charge will change.

Scheme of experiment 1.The electrostatic interaction of charges near the collision of masses could be measured, on the opposite sides of a shield stopping the moving body. For protection against electromagnetic radiation, a grounded screen should be placed between the shield and the device measuring charge interactions. In a similar way, the electrostatic interaction of charges can be measured near periodically moving masses, for example, near a rotating or linear moving neutral body.

Scheme of experiment 2.Electric current (pulses of electric current) is measured in a charged conductor when one end of the conductor is placed near a mass change, such as that generated by a rotating uncharged body. The other end is placed at some distance. Between the rotating body and the conductor, a screen is placed to protect against a possible electrons transfer induced by friction of the rotating body of air. The ideal case would be a rotating body in a vacuum chamber. Since the change of mass at a given point in space causes a change of density ρ0, i.e. a change of charge, different potentials will be generated at different ends of the conductor.

DEFLECTION OF LIGHT BY CHANGE OF MAGNETIC FIELD 

Background.Since magnetic field changes (as well as the gravity changes) are due to changes in the density of n=0-objects(I) ρ0, light should deviate from a straight-line movement not only in strong gravitational fields but also as a result of changes in strong magnetic fields.

Scheme of experiment.The deflection of a beam of light is measured near a pulsed electric current that creates strong magnetic pulses. A similar experiment was carried out by Eugene Podkletnov and gave results consistent with the proposed theory (Study of Light Interaction with Gravity Impulses and Measurements of the Speed of Gravity Impulses – Pp. 169-182 (14) Evgeny Podkletnov and Giovanni Modanese. Gravity-Superconductors Interactions: Theory and Experiment by Giovanni Modanese, Glen A. Robertson DOI: 10.2174/97816080539951120101; eISBN: 978-1-60805-399-5; 2012; ISBN: 978-1-60805-400-8)

CHANGE IN THE WEIGHT OF A BODY NEAR THE CHANGE OF MAGNETIC FIELD

Background. According to the proposed theory, the density of n=0‑objects(I), ρ0, is changed by moving electrons and positrons. As a result, a magnetic field is generated. Accordingly, a change in magnetic field strength leads to a change in the density of n=0‑objects(I), ρ0. The gravitational interaction is due to changes in density ρ0. For this reason, the local change of magnetic field will temporarily affect the gravitational attraction.

Scheme of experiment. It is proposed to produce unidirectional, pulsed, electric discharges. As protection against any electromagnetic radiation, a grounded screen should be installed. Behind the screen, the gravitational attraction is measured during the discharge. A similar experiment was carried out by Eugene Podkletnov and gave results consistent with the proposed theory (Evgeny Podkletnov, Giovanni Modanese. Impulse Gravity Generator Based on Charged YBa2Cu3O7-x Superconductor with Composite Crystal Structure, arXiv physics/0108005, 30.08.2001).

CHANGES IN THE WEIGHT OF A BODY NEAR OTHER MOVING BODIES

Background. According to the proposed theory, the density of n=0‑objects(I), ρ0, determines the gravitational attraction. This means that a change in density ρ0, resulting from the motion of electrons and positrons (as well as ionized and unionized atoms), will cause a change in gravitational attraction. The motion of electrically neutral bodies will also change the density ρ0.

Scheme of experiment. A dense body is accelerated to maximum speed and allowed to collide with the massive screen. Behind the screen the gravitational attraction is measured at the moment of the collision.

APPENDIX 1. CONSEQUENCES OF h‑SPACE THEORY BEYOND MODERN PHYSICS

TETRANEUTRON, DIPROTON, DINEUTRON, MULTI-MUONS

In the suggested theory, protons and neutrons are seen to be complexes of electrons and positrons. Unstable complexes having even more electrons and positrons are also possible, such as the tetraneutron (Sherrill, B.M. and Bertulani, C.A. 2004 Proton-tetraneutron elastic scattering Phys. Rev. C 69, 027601), the diproton (Raciti, G. et al., 2008 Experimental Evidence of 2He Decay from 18Ne Excited States Phys. Rev. Lett. 100, 195203–06) and the dineutron (Spyrou, A. et al., 2012 First Observation of Ground State Dineutron Decay: 16Be Phys. Rev. Lett. 108: 102501. doi:10.1103/PhysRevLett.108.102501) (see Fig. 14a). According to modern theoretical concepts, these complexes do not exist. Muons, in the suggested theory, have the same composition as protons, but in contrast to the proton, an unstable spatial configuration. Because of this, we can also consider as real the multi-muons, as unstable complexes, clusters of electrons and positrons. Multi-muons have been detected in collisions of protons and antiprotons (CDF Collaboration 2008 Study of multi-muon events produced in p-pbar collisions at sqrt(s)=1.96 TeV arXiv:0810.5357). Another unstable cluster can be a complex of three protons shown in figure 13a.

VARIABILITY OF CONSTANTS

In the proposed theory, the speed of light and the “Planck constant” are both constants. The charge of the electron corresponds to the density of n=0-objects(I) ρ0and, accordingly, decreases with time. This reduction in charge can explain the results obtained by astronomer John K. Webb, who in 1999, discovered that light from a distant quasar 12billion light years, is absorbed by metal ions in interstellar clouds, but the absorbed photons do not correspond to the spectra of the metals. Since the interaction of light with matter is determined by the fine-structure constant, α, then it has been suggested that α had a different value. This assumption is not consistent with modern physical concepts. All three constants, which determine the alpha (α = e2/hc) – the electron charge (e), the speed of light (c) and the Planck constant (h), cannot be changed. In the proposed theory, however, the electron charge is not constant. Accordingly, the alpha should decrease with time, offering an explanation for the observed change in the absorption/emission spectrum of metals.

BOUNDARY OF GRAVITY AND KUIPER BELT

In the proposed theory, the effect of gravity has a boundary that is determined by the density and size of a body. In the suggested theory, it is assumed that the Sun has a maximum density ρM , similar to the density of atomic nuclei. Under these conditions, the density of n=0‑objects(I) ρis maximal and equal to ρ0, ρM = ρ0 = 1010. The boundary of the Sun’s gravity can be calculated, taking into account the linear dimension of the Sun, and is ≈ 1013 m (see “Gravitational attraction”). This is comparable to the distance from the Sun to the Kuiper Belt, ≈ 1013 m.

To assess the limits of gravity of the planets (see “Gravitational attraction”), their density ρM is assumed to be less than that of the Sun. Then, the density of n=0-objects(I) ρis ≈ 106–107. The calculation for the Earth shows that the boundary of its gravitational action is around 10million kilometers.

ANTI-GRAVITY

The cause of gravity in the proposed theory is due to the reduction in density of n=0-objects(I) ρ0, or, more precisely, the change in the density of the vacuum ρM, around the body, as a result of the displacement of n=0‑objects(I) by the body. In addition to gravity, a change in the density of n=0-objects(I) ρ0takes place during the generation of magnetic field. It is characterized by the density ρΔ appearing as the result of electrons/positrons motion. Changes in the magnetic field, i.e. the changes in the density of n=0-objects(I) ρΔ, is associated with changes in the density of n=0‑objects(I) ρ0, and therefore should lead to a change in the gravitational attraction, which is associated with the decrease in the density ρ0.

COLD FUSION – LOW ENERGY NUCLEAR REACTION (LENR)

After the report in 1989 by Martin Fleischmann and Stanley Pons, the majority of cold fusion (LENR) experiments were performed by electrolysis of heavy water with a palladium cathode. The results (excess heat, helium generation,nuclear transmutations, neutron and tritium radiation) were inconsistent and contrary to the official current view on the synthesis of helium nuclei from deuteron. In another type of LENR experiment, nickel rods saturated with hydrogen were heated, and an excess heat was reported (Anomalous Heat Production in Ni-H Systems. 1994 S. Focardi, R. Habel and F. Piantelli, IL NUOVOCIMENTO VOL. 107A, N. 1; Large excess heat production in Ni-H systems. 1998 S. Focardi, V. Gabbani, V. Montalbano, F. Piantelli and S. Veronesi, IL NUOVO CIMENTO VOL. 111A, N. 11). In 2011, interest in LENR research was bolstered by the public demonstration of the E-cat device, performed by Andrea Rossi. In the E-cat, nickel powder is saturated with hydrogen and heated under high pressure, resulting in a significant excess of heat. The fuel in the E-cat contained additional elements besides the nickel, and according to the patent application, analysis of the fuel after its use in the E-cat showed that it contained a number of different elements, indicating that both nuclear fusion and fission had occurred during operation of the device (http://www.journal-of-nuclear-physics.com/files/Patent_WO-2009-125444.pdf). In 2013, a test of an improved design (E-cat HT) confirmed the previously observed thermal effects (Levi, G., et al., 2013 Indication of anomalous heat energy production in a reactor device arXiv:1305.3913), and in 2014 another independent test of Rossi’s device was performed (http://www.sifferkoll.se/sifferkoll/wp-content/uploads/2014/10/LuganoReportSubmit.pdf). The investigators detected lithium, aluminum and iron in the initial fuel. The heat excess was again confirmed and it was found out that nickel and lithium in the ash had different isotopic ratios in comparison with the initial fuel. Of particular significance was a decrease in the amount of 7Li, indicating the possibility of its fission.

In the proposed theory, LENR can be explained by the positron-electron composition of nuclei and the Coulomb law at atomic scales. According to the proposed theory:

  1. in atom, at distances between 1015to 1010 meters, there is no attraction/repulsion of electrons and positrons;
  2. in atom, attraction/repulsion increases from zero at 1010 meters to a maximum at 10meters from the nucleus;
  3. in the nucleus there is no strong interaction. Electrons and positrons are held by their attraction and repulsion to each other with a velocity of 10meters per second.

Thus, in the proposed theory the problem of overcoming the Coulomb repulsion between the nuclei and protons is not relevant for a range of distances from 1015 to 1010meters. In the range of distances 105–1010 meters the repulsion exists and decreases from 10meters to 1010 meters. The same decrease is true for the distance bigger than 105meters. In condense matter, in the range of 105–1010 meters, the repulsion between the nuclei and protons can be compensated by an attraction to the electrons placed at the same distance from the nucleus.

The heating of a proton-saturated nickel powder would increase the mobility of electrons and protons. Additionally, the pulsed magnetic field can also accelerate protons and electrons. The protons released from nickel powder can collide with themselves and lithium nuclei (as well as with nickel, aluminum and other elements nuclei). The protons in nickel lattice can collide with themselves, electrons and nickel nuclei. All these can lead to:

  1. the fission of proton and lithium nuclei (as well as nickel, aluminum and other elements nuclei);
  2. the fusion reactions generating neutrons and the nuclei not present initially, as the complexes of positrons and electrons.
  3. fission of nickel nuclei and nuclei of other elements.

The LEN fission reactions should be more effective at the distance range of micrometers since at the distances from 105to 1010 meters the Coulomb repulsion between the nuclei and protons decreases. This implies that nickel powder in size of microns can be favorable for effective LEN reactions. For this distance in space between the nickel particles, the colliding protons and lithium nuclei will have the decreasing Coulomb repulsion from maximum to zero. If the distance between the protons and lithium nuclei is bigger than 10meters, their collisions will face the opposite effect, the increase of the Coulomb repulsion having maximum at 10meters. This makes less probable the collision of the protons and lithium nuclei, followed by their fission, from the distance bigger than 10meters.

All fission reactions will generate heat and nuclear transmutations. The high level of heat production seen in the E-cat is due to the fission of lithium due to the interaction of lithium nuclei with protons, resulting in synthesis of beryllium, 84Be, followed by the 84Be fission to two alpha particles. The energy of generated alpha particles can be consumed further by protons, and this will keep the LEN reactions in self sustain mode, without external heating. The low efficiency of the earlier experiments with nickel rods can be explained by a lower rate of protons and nickel nuclei fission in comparison with E-cat having lithium as fuel.

In the proposed theory, nuclear fusion is possible as secondary reactions, but they will not produce the heat excess in LENR devices. The claimed thermonuclear effect of deuterium fusion in the thermonuclear explosion can be interpreted as the lithium fission since lithium (6Li) deuteride fuel was used in this process.

FORMATION OF FOUR-DIMENSIONAL SPACE

In the proposed theory, after a while three-dimensional space is predicted to be replaced by four-dimensional space. The important question then is when this can happen. To answer this it is necessary to compare the length of objects of four-dimensional space with the current linear dimension of the universe. Today, the linear size of the universe is estimated at ≈ 1026 m. With the creation of four-dimensional space, the length of objects of four-dimensional space, and the total length of the universe in absolute length units is 1030. Since the absolute unit of length is equal to ≈ 105 m, it follows that the current size of the universe is close to the total length of the beginning of four-dimensional space. Thus, the change from three-dimensional space to four-dimensional space could be very soon. Another estimate of the time of transition can be obtained by assuming the identity of the constant = 1/√ε0μand the velocity v0ρ0. The transition to four-dimensional space, as has been stated above, can happen at a linear size of the universe equal to the length of n=4-object. In this case, the velocity v0ρwill be reduced to the value equal to the velocity of one-dimensional objects, i.e. to the speed of light. A comparison of the values of the constant (velocity v0ρ0) and the speed of light can provide a rough estimation of the transition time. The constant cis the inverse square root of the product of the vacuum permeability, μ0, and the vacuum permittivity, ε0. In this product, the value of μis accurate and does not require the definition from the experiment. On the contrary εis determined from experiment.

Changes that will accompany the emergence of four-dimensional space, as noted in the section “Electrostatics”, are the change of electrostatic interaction from an inverse dependence on the square of the distance to the third degree of distance. This will lead to the destruction of atoms, which can be later regenerated. Electrons will be located at closer positions in the new atoms. Accordingly, the gravitational attraction will also change its dependence on distance, from the inverse square of distance to the inverse third degree of distance.

CHARGE CLUSTERS

The phenomenon of charge clusters was studied by Ken Shoulders (http://www.rexresearch.com/ev/ev.htm;http://www.svn.net/krscfs/Charge%20Clusters%20In%20Action.pdf). These clusters have sizes in the micrometer range, and have an excess of electrons in the ratio of the order of one positive ion for 100,000 electrons. The number of electrons in the cluster is 108–1011. Stability of the clusters, and the absence of repulsion between the electrons in them, has no explanation in modern physics. In the proposed theory, the size and stability of the clusters can be explained by electrostatic interaction at a distance of less than 105 m (see “Electrostatics” and “Atoms and Spectra”). The number of electrons also fits with the size of clusters. The linear arrangement of electrons (having a linear size 1015 m), when numbering 108–1011, correlates with the cluster size, of around 106 m. If the cluster size becomes larger, i.e. the distance between the electrons increases, this will change the nature of the electrostatic interaction between electrons, and the cluster will not be sustainable.