Gravity Beyond Einstein? Part II: Fundamental Physical Principles, Number Systems, Novel Groups, Dark Energy, and Dark Matter, MOND

3.1 LHC Data and New Particles

The Higgs particle, proposed in 1964, has been the only new elementary particle found by the LHC in Geneva (as of winter 2018). On December 3, 2018, the LHC was shut down to increase its luminosity and is planned to reopen in early 2021 [26]. In the meantime, the large amount of data the LHC experiments have already produced will be analyzed in order to extend our knowledge of fundamental physics. In particular, no elementary particle has been identified that could serve as a DE or DM particle. This is despite a recent extensive experimental search program at the LHC [27] confirming the nil results of many other experiments, already started some 40 years ago.

As was expressively stated by string theorists prior to the inauguration of the LHC, the LHC would be used in producing DM particles and observing their properties. Thus, the LHC combined with SUSY was supposed to be the key to understanding DM. The lack of DM particles suggests that they likely cannot be produced in the LHC. However, the gravitational impact of DM has clearly been measured (e.g. weak gravitational lensing), which means either DM is not composed of particles or DM particles do not exist in our spacetime (see below). In agreement with the results of the measured valid range of Newton’s law [6], the updated LHC did not find any energy leaks at length scales of ≈ 10⁻¹⁹ m, further rendering improbable the concept of extra higher space dimensions – which is fundamental to all types of string theories.

Most recently (2017), intensive search for WIMPs has taken place, following the upgrade of the LHC to s=13 TeV (centre of mass energy) in 2015. In the particle models beyond the SM, it is assumed that WIMPs may be pair-produced in collider experiments by means of hitherto unknown fields (bosons) by a coupling of DM particle(s) and the baryonic matter of the SM. If any WIMPs were produced in a collision, then by their very nature they would escape detection. However, a transverse momentum deficit should be detectable, yet the CMS collaboration [28] has been searching for this type of event for several years without finding anything. The results of this search will be discussed further in Part III, where a comparison with self-interacting DM models is made.

Der Intellekt hat ein scharfes Auge für Methoden und Werkzeuge, aber er ist blind gegen Ziele und Werte.

Albert Einstein

4.1 Formulation of Cosmic Physical Principles

Following Einstein’s idea, the following set of fundamental or cosmic principles are used as the core for any advanced physical theory, prior to its mathematical formulation. Of course, it is unknown whether this set is complete. These principles serve as blueprints (or guidelines) of the mysteries in an otherwise inscrutable universe. The validity of the chosen set of principles can only be justified in their success in describing physical reality. Establishing a basis of principles should arguably be the first step in establishing any comprehensive theory of space and matter. This approach is as close as possible to an axiomatic formulation of physics.

1. Principle of duality: It governs all physical events in the cosmos as well as the emergence of the cosmos itself. The cosmos, that is, order, requires from its very inception the duality of space and matter.^[15] It is considered (in EHT) the key principle of physics. In particular, duality requires that formation or creation must be followed by annihilation; i.e. the cosmos itself eventually will be annihilated. In particular, the two primeval, independent entities are geometry and energy. They are interdependent and are to be considered as two sides of one and the same coin. One state is as much a part of its opposite state as are the two sides of a fabric. This geometry–energy duality requires the spacetime lattice (composed of elemental surfaces with spin) and the associated primordial energy, namely, DE, to be generated simultaneously (see also [39] the discussion of the double helix).

   In quantum mechanics, duality appears in the form of the wave–particle duality governed by Heisenberg’s uncertainty relation. Nevertheless, matter always comes in the form of particles.

   It is important to note that the duality principle strictly limits the reductionist viewpoint, namely, the belief that all interactions can be unified. First, there exists the duality of geometry (the stage) and energy/matter (the actors) that according to this principle cannot be unified. This means that Einstein’s GR (cosmological gravitational fields) and quantum theory may not be unified in the form of quantum gravity as previously thought and sought, for instance, as sought, see C. Kiefer in Chapter 5 in [40] and also in his more recent monograph [41]. As will be outlined in Section 5.3 the grid spacing of the space lattice may not be the Planck length but could be many orders of magnitude smaller. This means that the Planck length ℓ_Pl may not be the proper length scale for the spacetime lattice.

   Second, the mathematical groups discussed in Section 5.4 predict the existence of a second kind of gravity, represented by three additional gravitational bosons, which should result from the interaction of gravity and electromagnetism. This interaction is termed hypercomplex-gravity fields, which would be many orders of magnitude stronger than the cosmological gravitational fields. It is these four fundamental interactions, characterised by the three constants

   GN,c,ℏ,

   to which the unification process should be applied.

   GR is a classical theory because in its formulation neither the wave-duality picture is employed, nor is Planck’s constant ℏ needed. Hence, the Planck length,

   ℓPl=GNℏc3=10−35m,

   constructed from GN,c,ℏ, may not be the appropriate length scale for the spacetime lattice. By contrast, the grid spacing of the spacetime lattice may be considered to be governed by the constants

   GN,c,mp,

   (m_p denotes the mass of the stable proton), as will be further discussed below when the principle of dual energy is addressed. With regard to a length scale of gravity, it seems to be more natural to consider the length obtained when the gravitational self-energy of a particle is equal to mc2 as the proper grid spacing Δs of the spacetime grid. In particular, as the proton is the most fundamental stable baryon, one obtains for the grid spacing of the spacetime lattice

   GNmp2Δs=mpc2

   or

   (1)Δs=GNmpc2≈10−10 10−27(3×108)2≈10−54m,

   resulting in a length scale about 19 orders of magnitude smaller than the Planck length, indicating an almost completely rigid grid untouched by almost all comprehensible energetic phenomena. If this were the case, then strong gravity [42], as obtained from the world of particle physics, that requires

   (G⋅mc2)2=1L2

   would mean that the curvature of spacetime has to be compatible with this length scale, L, far below the Planck length ℓPl, and consequently, effects from quantum gravity would not be visible. Furthermore, ℓ is clearly not on the TeV scale as recent LHC data have shown. Thus, gravity could be considered weak in all circumstances. Compared to the Compton wavelength of the electron ƛC=ℏmc≈3.86×10−13 m, Δs is an extremely small number. On the other hand, if one assumes that the universe contains about 10⁸⁰ nucleons, the radius of the universe is determined as R_U ≈ 10²⁶ m. With a grid spacing of 10⁻⁵⁴ m, spacetime can be regarded as a manifold with regard to all energetic aspects conceivable. If this length scale is converted into mass, one obtains the Schwarzschild energy scale M_S ≈ 10³⁴m_p, and new physics should appear. Such a particle should be totally unstable and immediately decay because of its humongous mass, creating a massive explosion.

2. Principle of extra systems of numbers: Nature uses additional systems of numbers to produce different types of matter (or different flavours of matter) and to process information. Number systems employed in current physics are the rational numbers ℚ, real numbers ℝ, and complex numbers ℂ. In addition, nature may utilise also hypercomplex numbers (quaternions) ℍ as well as octonions 𝕆 (Section 5.4). Extra systems of numbers then are used, replacing the concept of extra space dimensions. Extra number systems have major ramifications in physics. For instance, they may lead to different symmetries and larger groups compared to the SM of particle physics. In particular, additional gravitational bosons are predicted, leading to extreme gravitational fields outside GR. Such bosons may provide the enabling technology for gravitational engineering and propellantless space propulsion, pursued by space industry and NASA since the 1960s (Section 8).

3. Principle of optimisation: This means that nature is perfectly optimised within the framework of existing physical laws. If only a single bit of information would be removed from the total information content of the cosmos, it would break down. Nature does not use trial and error processes in her physical laws. Instead, all of her processes take place such that no process can be found that would result in a higher degree of optimisation (e.g. Carnot engine). In this regard, there is absolutely no redundancy in nature. Mathematically, this principle can be expressed by the Hamiltonian (or variational) formalism that is applicable to both classical and quantum physics (see below). This principle was already employed in the 17th century by the French mathematician Pierre de Fermat to set up a universal law for the propagation of light. This principle also requires that gravity has to be derived from an invariant action principle as is the case for GR (D. Hilbert, 1915).

4. Principle of quantisation: Nature does bookkeeping (accounting and counting), i.e. uses (half) integers only. In reality, there are no continuous physical quantities, only discrete quantities. This means infinities or singularities cannot exist in physics. In addition, spacetime ultimately has to be a lattice, which in principle should have an effect on the propagation speed of light waves with respect to their angular frequency ω, although such an effect would be extremely small (see the remarks on the ESA Integral satellite measurements Section 5.3).

5. Principle of quantum fluctuations: Nature does not know static physical states. Even in the case of zero external energy, there exists constant interaction with the virtual particles of the vacuum (i.e. the state with no real particles) of spacetime. This manifests itself in the form of quantum fluctuations, governed by the Heisenberg uncertainty relation and the duality principle.

6. Principle of finite existence time: All objects (material or immaterial) as well as structures (e.g. spacetime itself) in the cosmos (universe) have a finite time of existence because of duality and quantum fluctuations. This principle is not independent of the other principles, but because of its great importance, it is listed separately. This also includes the cosmos itself. In order for something to exist, the cosmos must exist in a dynamical but time-limited state. For example, directed motion is ubiquitous, but nothing can inflate to eternity.

7. Principle of causality: There exists an arrow of time (realised by ubiquitous motion of physical objects) putting the equivalence principle into action by governing the direction of all physical processes and the building of ever more complex structures (information content). Under limited circumstances, the arrow of time may be reversed by cosmic symmetry breaking, such as low cosmic background temperatures acting as a governing parameter by forming a Bose–Einstein condensate.

8. Principle of dual energies: In order to build a universe, both structure and matter are required. Structure reflects potentiality through admissible forms of organisation (information), while matter is employed to actualise its form. Hence, there must exist two fundamental types of energy (duality principle).

   First, there is the relation between energy and information, that is experimentally verified, i.e. energy of information that is connected with the name of L. Szilárd, 1929,

   (2)E=nbit 0.69κBTln2,

   where n_bit is the number of bits comprising the total information content of the physical system (i.e. the crystal like space lattice), and T denotes temperature. A single bit of information is equivalent to the amount of 0.69κBT of energy. The same amount is necessary for deleting one bit of information. We this call Szilárd’s equivalence principle of energy and information. It should be mentioned that an experimental value of 0.28κBTln2 has been established (see Chapter 9.7 in [9] that discusses an Ising model of spacetime) along with the concept of entropy^[16]

   S=κBlnΩ

   where Ω denotes the number of microstates, and the assumption of an adiabatic expansion process of the universe, the classical formula

   dS=1TdE

   can be used to define the concept of temperature that has to be utilised in (2). It is evident that the universe must have started with the lowest entropy possible, comprising two elemental surfaces only – the smallest universe conceivable. There are only two (spin) orientations possible, parallel (↑↑) or antiparallel (↑↓). The (negative) potential energy resulting from the spin interaction of the first two metrons may be described by an Ising model. Energy conservation then is satisfied by the production of DE (particles) that are of positive energy density. That is, the initial entropy of this universe is given by

   S=κBln2.

   The inflationary evolution of the space grid describes the rapid creation of elemental surfaces, driven by the generation of DE (and vice versa). This leads to exponential growth in the number of microstates Ω, accompanied by the elevation of temperature T. Hence, the negative energy of information, (2), has to be exactly compensated by the positive energy of matter, (3), as a consequence of universal energy conservation. This, in a nutshell, is the basic idea of how our universe might have started from nothing. Hence, it is described by the term Quantised Bang.

   Second, there is the energy of matter

   (3)E=mc2=ℏω

   that describes material particles with rest mass m or radiation of angular frequency ω, respectively. It is synonymous with the name of A. Einstein and his historic articles published in 1905 and 1906. The two types of energy are assumed to have been converted into each other during the inflationary phase of the universe and therefore are regarded dual to each other. Moreover, if the universe enters into its contraction phase, marked by a reduction in the number of elemental surfaces, DE necessarily diminishes as well and with it all kinds of matter.

9. Principle of energy conservation: Energy is the physical quantity that produces changes in the states of physical systems. The total energy in the universe is

   EU≡0.

   In conjunction with the duality principle, this energy may be split into negative (atoms of space) and positive (DE) energies. Due to the optimisation principle, energy is conserved. Owing to the quantisation principle, energy comes in discrete quantities only. Because of the principle of quantum fluctuations, energy change ΔE during a period of time Δt obeys the Heisenberg uncertainty principle,

   ΔEΔt≥ℏ.

10. Principle of dual universe: According to the renowned British physicist R. Penrose [43], there exists a mathematical universe (platonic world) for the blueprints of coupling constants and physical laws (cosmological principles) beside the physical universe that enacts their actual implementation in space and time by means of material particles. This means coupling constants cannot be derived from physical theory.

11. Principle of non–self-interaction: Any physical system that comprises a single system cannot interact with itself (Fig. 3). In particular, a self-force does not exist because it is based on the concept of retardation owing to special relativity (SR), which is not applicable to a single physical system.^[17] Otherwise, there would be a contradiction, because if there is a single physical entity only, nothing exists to interact with. For instance, this means that an electron e⁻ cannot interact with its own field as assumed in classical electrodynamics for there is no second partner available for physical interaction. In particular, a self-force does not exist, because it is based on the concept of retardation owing to SR, which is not applicable to a single physical system. However, in general the concept of a single entity system will depend on the type of the specific interaction but is independent on spatial separation and on the number of particles. As long as the system can be described by a single wave function with regard to the physical phenomenon considered, e.g. two entangled photons, such a system nevertheless comprises a single entity, to which the concept of space and time is no longer applicable (see below). This means that, for instance, the spin of two entangled photons must be considered as a single quantity; i.e. two independent photon spins do not exist. In case the description by a single wave function ceases to be valid (because of decoherence) (interaction with the environment that destroys entanglement, e.g. exchange of energy), the system loses its single identity and separates into distinct parts, now capable of interacting with each other.

12. Principle of organisation: The duality principle not only requires dual energies but also demands a dual to entropy, which is organisation. This principle is at work everywhere and is enabled by energy. With regard to matter, it is at work forming the entire material hierarchy, combining primordial gluons and quarks to elementary particles like the nucleons, proton, and neutron. Nucleons are then assembled into more and more complex atoms that in turn are employed in the formation of molecules. Eventually, organic molecules are built from carbon chemistry and are finally assembled into polypeptides that are at the foundation of life itself. The next step in this chain is the formation of biological cells, already at a mass of about two nanograms, comprising millions of proteins, which are the basis of all living structures, which distinguish themselves by ever increasing organisation levels. Although energy is necessary in the formation of complex structures, it is the organisation principle that governs and directs the evolution of most complex biological structures, clearly demonstrating the existence of Aristotle’s ancient concept of entelecheia,^[18] that is, demonstrating that this universe has a clear objective, namely, providing the basis for life. Life is not a process that started accidentally; that is, life is not the result of a trial and error process. The complexity of the human body is so enormous that it is totally impossible to build such a structure by pure chance, hence the entelechial and aeonic dimensions. The associated probability is zero; therefore, the definition given by NASA that “life is an autonomous chemical system that is capable to follow Darwinian evolution” is far too limited and provides zero chance for life to exist. The picture portrayed by the late B. Heim [31], who stated that the process of life proceeds in phase transitions, i.e. abrupt changes (steered by the internal entelechial dimension), followed by a longer era of evolution (aeonic dimension), appears to be more logical than the 19th-century Darwinian view that covers only a very limited section of the entire cycle of life and thus arrives at insufficient conclusions.

Figure 3:

A physical system can interact only if it comprised separate parts and its two wave functions do not overlap (upper picture, a). In other words, an individual physical system cannot interact with itself; i.e. there is only one wave function (lower picture, b). For instance, the electron as an elementary particle cannot interact with its field. Particle and field are two different but equivalent ways in describing the same single physical object, namely, the electron.

4.2 Cosmic Principles and No-Go Theorems

Some of the most stringent physical consequences of the cosmic principles shall be outlined below. It will be shown that several essential physical concepts pursued for decades are no longer tenable within the framework of these fundamental principles.

A number of these constraints have been formulated in the form of no-go theorems to emphasise the limits on the physical features of the universe imposed. The cosmic principles impose severe and unforeseen, but far-reaching restrictions on any theory [9].

1. No-Go Theorem: Geometrisation of Physics: The geometrisation of physics as pursued by Einstein, Schrödinger, Wheeler, and, most recently, in the form of superstring theory is not compatible with the principle of duality (see below) and thus cannot succeed regardless of the level of mathematical sophistication.

2. No-Go Theorem: Unification of Physical Interactions: The duality principle excludes the complete unification of all physical interactions. In particular, it does exclude the conversion of fermions into bosons and vice versa via some kind of supercharge. Without any further mathematical investigation, the duality principle is ruling out any unification by the so-called supertheories. That said, duality must be a global symmetry that remains unbroken. So far, no deviation from this rule has been observed. This feature seems to be inherent to all physical phenomena.^[19]

   With respect to unification, Maxwell’s theory of electromagnetism was unified with the weak interaction. It should be possible to unify the strong force with gravity, which means hypercomplex gravity, but not with Einstein’s GR, which rests on the curvature of the space lattice and not on mediator bosons. However, it should not be possible to further unify all four interactions. This may be one of the rare occasions where A. Einstein’s intuition turned out not to be correct. However, as surmised by Faraday and Einstein, an interaction between electromagnetism and gravity may be possible, providing the cause for the postulated hypercomplex-gravity field. However, such a field is not part of GR and exists as a second, independent type of gravity.

3. No-Go Theorem: No Majorana Fermions in Nature: It is interesting that, according to the principle of duality, Majorana spinors should not exist. These fermions should annihilate each other as they are their own antiparticles. Also, according to the Feynman–Stueckelberg interpretation, antiparticles go backward in time, and thus, a Majorana neutrino must both go forward and backward in time. As there is no known physical mechanism for a neutrino to select the direction of time, this is considered a dubious behaviour. Hence, the ongoing GERDA experiment [44] should not succeed in finding any Majorana neutrinos.

4. No-Go Theorem: No Big Bang: The fact that the energy of the universe E_U ≡ 0 at all times excludes the occurrence of a Big Bang event. Energy is globally conserved at all times. Additionally, singularities do not exist in physics. Instead, the initial formation of the space lattice starts when two metrons are created by quantum fluctuations (Quantised Bang), and their spin interaction produces the first DE particle (owing to energy conservation). This is a self-accelerating process that finally comes to a halt when DE is converted into matter.

5. No-Go Theorem: No Infinite Universe: The fact that there are no infinities in physics means that the existence time for the universe is finite. Thus, there is the requirement that the current expansion era will have to come to an end, and a turning point in the cosmic motion will be reached, followed by an epoch of contraction.

6. No-Go Theorem: No Open Universe: For the very same reason the universe cannot be open, it must be closed or cyclic.

7. No-Go Theorem: No Wormholes: Wormholes cannot exist in the cosmos, because nature does not allow for singularities. Wormholes are admissible mathematical solutions of Einstein’s field equations but are not realised by nature. They are simply an expression of the infinite mathematical continuity of the equations that become invalid when the length scale considered is of the same order of magnitude as the grid spacing of the spacetime lattice, and the equations break down.

8. No-Go Theorem: Feynman Diagrams: Suppression of virtual particles, that is, in a physical process there is only a finite number of particles that are off-shelf, i.e. pμ2≠mc2. Instead of equations, Feynman used diagrams to account for all probability amplitudes that can occur in a scattering process, for instance, in the scattering of two electrons. In the formulation of quantum electrodynamics (QED), comprising the equations of Maxwell and Dirac that are both linear hyperbolic partial differential equations, a mathematical solution is obtained utilising Green’s functions in combination with an iterative procedure. Mathematically, such a scheme generates an infinite number of terms, but nature will stop the generation of an infinite number of virtual particles by virtual particles as required by Feynman diagrams, etc., once a certain nesting depth of the loops has been reached (far above the Planck length).

9. No-Interaction Theorem: Single Entity Physical System: Interactions within any physical system will cease if the information that permits to identify at least two distinguishable parts in this system is destroyed by changing the governing physical parameters, e.g. lowering the temperature (see above, the principle of non–self-interaction). A many-particle system (two or more) can no longer interact if being reduced to a single particle system. Thus, there is a major difference between leptons and hadrons. It should be noted that the concept of single entity system depends on the type of interaction. Two entangled photons form a single entity system with respect to their spin. A single entity system responds as a whole, which means that if a physical field were involved, retardation effects do not exist anymore. For instance, the electrodynamic potential A is no longer retarded by the time |x−x′|/c required for a light wave to propagate from point x to x′, i.e. concepts of time, distance, and propagation speed are not applicable to single entity systems. Thus, such a system is no longer subject to Einstein’s theory of SR. It is most remarkable that nature has realised such single entity systems.

These relatively simple fundamental principles, nevertheless, have produced far-reaching and surprising consequences for physical theories in general and also for the type and topology of admissible universes as will be discussed in greater detail in the next section.

4.3 Cosmic Principles and Gravitation

In particular, the field equations of GR may be directly deduced from the duality principle in conjunction with the assumptions that the total energy of the universe EU(t)≡0 at all times, conserving total energy.

In the following, several additional physical constraints resulting from the fundamental principles are elucidated.

In 1948, Feynman published a third formulation of quantum mechanics based on the concept of the propagator D (or K). His approach has the great advantage that the interaction of two particles (calculating the scattering cross section) can be pictorially described by an infinite set of diagrams (attributed to an infinite series obtained by an iteration process) that became known as Feynman diagrams (a modern discussion, for instance, is given by A. Zee [45]). Nothing can be said about the convergence of this infinite series in general, nor should it be taken for granted that each term in the series does have a physical meaning. As it turns out, the recent works of Z. Bern [46], [47] and A. Zee, Part. N in [45], show that this is actually not the case!

In certain physical situations, the principle of finite phenomena may be at odds with the general picture of Feynman diagrams, because these diagrams rely on the generation of an infinite number of virtual particles, e.g. coming from loops within loops, etc. Although it is claimed that Feynman’s approach has led to the most accurate computation in physics, viz. the calculation of the magnetic moment μ_e of the electron, this picture may be incorrect in certain situations. There are numerous physical situations – as was clearly revealed recently by Z. Bern and others – where his approach is a complete failure and leads to divergent integrals for the probability amplitude. In particular, this is true in the process of the quantisation of linearised gravity assuming a flat background Minkowski metric

where hμ⁢ν describes the (small) deviation from the chosen background metric. This approach is fully divergent, and the problem seems to lie in the accumulation of an infinite set of (physically nonexisting) Feynman diagrams producing (nonexistent) off-shelf or virtual particles (on-shelf in this context denotes real physical particles of the SM).

A similar problem occurs in the calculation of the classical multipole radiation of order ℓ for which the average radiation power Pℓ (averaged over the surface of a sphere), valid for both magnetic and electric multipole radiation, is given by the expression

with kr0≪ℓ, where k is the wave number and r₀ denotes the linear dimension of the radiator, e.g. an electron oscillating in an electromagnetic field of an atom. As there is no mathematical limit for ℓ, this expression would lead to Pℓ→∞ for ℓ→∞. In electrodynamics, this is not considered a problem, because, e.g. for light one can choose k = 10⁷ m⁻¹ and r₀ = 10⁻¹⁰ m for an atom. Hence, electric quadrupole radiation (ℓ = 2, so-called forbidden spectral lines in atoms, but observed in the interstellar gas) is suppressed by the factor (kr0)2×4×10−4=4×10−8 compared to dipole radiation (ℓ = 1). On the other hand, if an electron radius r₀ = 10⁻²⁰ m (LHC limit) is assumed and ℓ = 10⁶, then k = 10²⁴ m⁻¹ is an admissible value and results in the factor

which obviously is a nonphysical value for the multipole radiation power P_ℓ. Of course, such a value has never been observed, and it is physically totally meaningless, because the linear dimensions of the radiators have to be taken into account. In general, if these dimensions are not too large, only a few multipole fields are needed that are orthogonal to the prevailing dipole fields. It can be expected that something similar holds true for Feynman diagrams. The mathematical solutions of the spherical wave equation, according to which any electromagnetic field can be expanded, form an infinite series. However, only a few terms of this series are utilised by nature because of the inherent length scale limit for the radiators. Hence, nature does not produce any divergences, but the mathematical solution does, for it knows nothing about limits of k or r₀, nor ℓ. In Feynman diagrams, it can be expected that the generation of virtual particles by virtual particles that were generated by virtual particles and so on is suppressed in a similar way.

The principle of distinguishability means that only distinguishable physical systems can physically interact; e.g. only two particles that can be distinguished by experimental means can physically interact. This question is not as innocent as it may sound because the particles must be close enough together so that their probability density amplitudes overlap significantly – whatever that may mean in practice. An important consequence of this principle is that as soon as the wave functions of individual particles can no longer be separated, they form a single physical system, and the concept of locality is no longer applicable. Einstein’s SR is not valid for such a system. Consequently, physical interactions should diminish depending upon the extent of the overlap and eventually disappear. Particles must be identifiable (information content) as independent physical systems in order to be able to interact with each other.

A single physical system cannot interact with itself. For instance, this may be exemplified regarding the electron. It is incorrect to consider an electron as an entity comprising two separate parts, namely, a particle and a field at the same time. This means that formulating the electron Hamiltonian dynamics via the classical action describing two features – the action function for a free particle of rest mass mc² and a second action resulting from the charge q of the electron and its own field given by the four-vector A(x) – necessarily leads to a contradiction. In order to describe this particle-field system, the action of the free electromagnetic field, specified by the field tensor Fμ⁢ν, must also be added. The total action S can be regarded as a functional of the variables x_μ and the field components A_μ. The field components are determined from the charge q of the moving electron acting as the source (RHS) j_μ in the d’Alembert equation for the A_μ. However, the equations of motion resulting from this formulation are nonphysical because the motion of the electron is self-accelerated, producing an infinite self-acceleration over time. Although mathematical techniques in QED have been devised to avoid this kind of trouble, the problem is with the nonphysical and logically contradictory assumption that a physical system, which is to be understood as a single entity, does interact with itself. It should be remembered that in quantum mechanics either the particle or the field picture is applicable, but not both at the same time. Hence, in the case of the electron, the formulation of the action S utilising three different terms is physically incorrect. Born and Infeld have tried to remedy this problem by adding nonlinear terms to the Lagrangian, but this comes at a cost of, for instance, a charge density ρ ∼ r⁻¹³.

By contrast, if the spatial extent of the wave functions of two physical systems is increased by changing external physical parameters so that their wave functions overlap (e.g. by lowering the temperature), and finally only a single indistinguishable system remains, then this resulting single entity no longer exhibits any physical interaction. Such a system is a single system, and no information is available to identify two different parts incapable of physically interacting with itself.

This may lead to dramatic consequences that should be experimentally observable. The Compton or de Broglie) wavelength of a particle is given by λp≪λe and λ=ℏ/mc. Two particles are distinguishable if their spatial distance (separation) d≫λp. The de Broglie wavelength λ_p strongly depends on temperature T. Hence, a physical system may become indistinguishable with lower temperature. This fact will be exploited in an attempt to explain the systematically different values measured of Newton’s constant G_N that will be dealt with in Part III. One class of experiments is performed at room temperature utilising torsion balances, whereas a second class of experiments uses ultracold ⁸⁷Rb rubidium atoms (atomic number 37) at 1.8 mK. A further important consequence of the non–self-interaction principle may be the suppression of the generation of virtual particles in Feynman diagrams.

Physics without strings is roughly analogous to mathematics without complex numbers.

Edward Witten, October 1998, on the importance of number systems in physics [48]:

If supersymmetry plays the role in physics that we suspect it does, then it is very likely to be discovered by the Large Hadron Collider (LHC), or its upgrades, at CERN in Geneva, Switzerland.

Edward Witten [49], November 20, 2012

5.1 Hermetry Forms

In the 1925 theory by Kaluza, it is necessary to construct a 5 × 5 metric tensor g_μν including the components A = (A^μ) of the electrodynamic potential in order to unify gravitation and electrodynamics (although this was later proved to be incorrect by Jordan in 1949, see Part I). That is, at each point in spacetime, a circle is attached (i.e. the fourth space dimension is curled up and thus is not directly visible) but is considered a real space dimension. In string theory, six (or more) curled-up space dimensions are added that lead to a 10-dimensional metric tensor aimed at describing gravitation, electrodynamics, and strong and weak forces, as well as quarks and leptons (matter). Such a metric is termed monometric. However, as was shown in the previous section, this approach is not supported by experiment.

The alternative physical model presented in this article, termed extended Heim theory or EHT (not to be confused with the theory of the late B. Heim, see our remark in Part I), employs an internal gauge space of eight dimensions, termed Heim space (𝖧8) that gives rise to a polymetric tensor comprising a set of 15 (hence poly) metric subtensors represented by 4 × 4 matrices. A metric subtensor gμν(Hℓ),ℓ=1,…,15 ([9] Chapter 5.2) is called a hermetry form. The term H_ℓ specifies the selection of internal coordinates ξ^a out of which the specific hermetry form is to be constructed. Hermetry is a combination of the two Greek words hermeneutics and geometry, denoting metric subtensors that have physical meaning, a term coined by B. Heim. It should be noted, however, that hermetry forms are obtained from the internal geometry of 𝖧8 which is the gauge space of EHT. Each hermetry form stands for a family of particles or fields and by utilising the complete set of hermetry forms a classification scheme for all physical interactions and particles. Hermetry forms are associated with groups as discussed in Section 5.4 (see also [9] Chapter 9.2 and [50], [51]). Employing the field of hypercomplex numbers (known also as quaternions) results in the so-called cosmic group O(32, ℍ) (see below). As one of the major consequences of this group, a total of six gravitational bosons are predicted. The cosmological gravitational fields as described by GR, EGN (static mass), and BGN (moving mass, analogous to electromagnetism) and their interaction with matter as well as DE are (only formally) characterised by three gravitational bosons. In addition, the postulated interaction of electromagnetism and gravity (based on the phenomenon of symmetry breaking) predicts the existence of extreme gravitomagnetic fields or hypercomplex-gravity fields, which may be generated in the laboratory, similar to the generation of magnetic fields. Hence, the source of these fields is entirely different from GR; that is, these fields cannot be due to the presence of large static or moving masses.

The eight-dimensional gauge space 𝖧8 is a fibre bundle attached to each four-dimensional spacetime point x (external space) and comprises four subspaces that determine the associated symmetries and hence also the group structure. Internal coordinates ξa,a=1,…,8 are used to construct the set of 15 internal metric tensors – in analogy to GR. Hermetry forms describe physical properties of the respective particle families. According to EHT, the set of hermetry forms is meant to account for the total particle inventory of the cosmos. Obviously, this approach is different from the so-called geometrisation of physics that solely relies on spacetime geometry.

The four subspaces of 𝖧8 are as follows:

1. 𝖱3 with three internal coordinates ξ1,ξ2,ξ3 describing matter (i.e. only those hermetry forms that contain at least one of these coordinates describe material particles),

2. 𝖳1 with coordinate ξ⁴ for energy,

3. S²with the entelechial and aeonic dimensions, representing organisation, with coordinates ξ⁵, ξ⁶, respectively,

4. 𝖨2 with coordinates ξ7,ξ8 that enable the exchange of information among physical systems, expressed by Szilárd’s energy principle, (2).

The concept of Heim space with its subspace structure is reflected in the mathematical group structure of EHT as discussed in this section. In particular, the entelechial dimension, ξ⁵, can be interpreted as a measure of the quality of time varying organisational structures (inverse to entropy, e.g. formation of complex molecules or plant growth), while the aeonic dimension, ξ⁶, tends to steer these structures towards a dynamically stable state (an aeon denotes an indefinitely long yet finite period of time). From these four subspaces, the 15 hermetry forms (metric subtensors) are obtained, as outlined in detail in Section 5.2 of [9].

This approach provides a relationship between geometry and physics (in some ways as foreseen by Einstein), but the major difference is that hermetry forms are living in an internal gauge space and not in spacetime. The geometrisation of physics, i.e. reducing physics to pure external geometry as pursued by Einstein and Wheeler, does not seem to be achievable.

As the universe cannot assume a static state (which would be unstable), the current expansion of the universe eventually must reach a turning point, reversing its current mode from expansion to contraction.

In the subsequent discussion, the concept of hermetry forms is no longer needed, once the group structures have been established for both spacetime and matter, which follow from the subspace structure of 𝖧8.

5.2 Extra Number Systems versus Extra Space Dimensions

An extensive discussion was presented in Part I [2] cf. figure 1 concerning the physical likelihood of extra space dimensions. Substantial experimental evidence recently derived from diverse areas includes the validity of Newton’s law, the dipole moment of the electron, and, in particular, the latest LHC results (2018) that did not find any hint for the existence of these dimensions. On the contrary, based on these experimental data, it seems to be more likely to conclude that extra dimensions do not exist.

With the advent of gravitational wave astronomy by the LIGO and Virgo Scientific Collaboration, numerous studies have been performed to detect signatures of extra dimensions in gravitational waves, most recently by Andriot and Gómez [52]. From a logical point of view, it is clear that if these dimensions exist they must have some kind of impact on the amplitude or frequency, etc., of the observed gravitational wave pattern, and that should lead to a deviation from the predictions of Einstein’s GR – at least theoretically.

Figure 4:

According to the group O(3,q∈ℍ)H that stands for the 15 particle families represented by hermetry forms, there are (see also the discussion in Sections 2.1.4. and 5.2.4 [9]) four lepton families. The fourth particle, the χ particle, the (long sought) dark matter (DM) particle has a (negative) mass of some −80.7 GeV/c², while the corresponding DM neutrino, the ν_χ lepton has a mass of approximately −3.2 eV/c². The masses of these two particles are negative, owing to their hermetry forms. Particles with rest masses different from zero must contain the coordinates of subspace 𝖱3. So-called degenerate hermetry forms, which have no cross terms between subspace 𝖱3, while the other subspaces are present in this hermetry form, and are thus considered to represent particles of negative mass [54]. Furthermore, apart from particles of positive and negative masses, there should be a third type of particles that possess hypercomplex mass (see text) like, for instance, the imaginary electron eI− and quark q_I. Hypercomplex masses in form, for instance, of the imaginary proton, p_I, are assumed to play a central role in the enigma of the different neutron lifetimes, measured in the beam and bottle methods [55]. Their hermetry forms are similar to the ones for the real electron or real quark, but subspace coordinates 𝖱3 are missing. For this reason, those particles are assumed to be virtual particles. The physical meaning of hermetry forms is further discussed in Section 5.2.4 [9].

If these deviations existed, it appears implausible that they would only manifest in a modified gravitational wave pattern but not exhibit, for instance, in the form of a modified Newtonian law. Extra dimensions clearly must have an influence on the energy that goes in the higher dimensions, and any energy leakage should be detectable at the length scale of these higher dimensions as predicted by Arkani-Hamed et al. [53] in 1998. However, as it is evident from Figure 2 (see lower figure, updated from Part I), there is now firm experimental evidence that extra spatial dimensions cannot exist. As a direct consequence, the idea of SUSY appears no longer tenable, and an alternative concept has to replace it. In EHT, as was stressed in Part I and also in this article, the novel concept of extra systems of numbers is introduced, and as a result, physical entities may be described by groups utilising the field of hypercomplex (or quaternionic, q∈ℍ) numbers. This would also give additional meaning to the equation for the Planck mass (4) that now can have mass values m∈ℍ. In analogy to the concept of quark flavour, the novel concept of mass flavour is introduced.

For m∈ℍ, there are the following five solutions of (4), namely

where it is assumed that negative mass represents DM, and hypercomplex masses i m, j m, k m may be generated that belong to the category of virtual particles, which are always created in pairs, e.g. +i m and −i m, etc. These particles may be produced during a reaction, acting as some kind of catalyser, but owing to their ephemeral nature, they are off-shelf.

The Planck satellite data revealed that 68.3 % of matter is in the form of DE. Extended Heim theory describes DE as the nonluminous precursor of matter and as such is not contained in the elementary particle scheme. The 4.9 % of OM (visible matter) residing in our de Sitter spacetime dS^1,3 is composed of positive mass m > 0, whereas the 26.8 % of DM has negative mass m < 0 (fourth particle family, Fig. 4) and is located in dual spacetime DdS^1,3, and thus, these particles are not directly detectable by experiment. Masses 19 are denoted as hypercomplex or quaternionic masses (i.e. virtual particles that are predicted to be generated in the conversion process from electromagnetism to hypercomplex gravity). Furthermore, if Higgs boson(s) are composite particles [56], then hypercomplex particles might play a role in providing the building blocks for Higgs particles, similar to how quarks are confined in order to constitute baryonic matter.

In 2005, P. Mannheim [57] published a comprehensive review of conformal gravity, a proposed alternative theory of gravitation to GR, in which he claimed that CG reproduces galactic star rotation curves without DM and DE, also mimicking the observed accelerated expansion of the universe. As we now know, DM does exist as is revealed from the collisions of galaxy clusters, e.g. bullet cluster, see figures 9.18–9.20 in [58], [59] that clearly illustrate the DM distribution, and these claims therefore must be incorrect. Hence, CG cannot be a viable alternative to Einstein’s GR.

5.3 Lagrangians, Symmetries, and Groups

The question naturally arises whether the cosmic principles formulated in Section 4 can be quantified sufficiently to provide the guidelines in constructing the mathematical groups representing the complete inventory of the physical particles comprising all types of matter existing in the universe. Of course, one could select a group, for instance, such as SU(5) or SO(10) (unification unitary groups are needed to conserve probability density), and based on the number of their generators (n² − 1) for the SU(n) group and 12n(n−1) for the SO(n) group, one could try to identify the proper particles. However, this approach has been tried for decades and proved not to be successful. A modified procedure will be presented below in an attempt to describe the (complete) particle inventory of the universe including the spacetime lattice itself. One of the results of this approach is the prediction of the additional hypercomplex-gravity fields, mediated by two additional gravitational bosons of spin 1 and one spin 0 boson.

M. Faraday (1791–1867), one the most influential experimental scientists in the field of electrodynamics in the 19th century, was the first to introduce the concept of electric field E and magnetic field B. The field concept turned out to be instrumental in the formulation of electrodynamics by James Clerk Maxwell in 1864. Several decades later in Einstein’s special and general theory of relativity, Faraday’s field concept was extended to spacetime itself (also termed the vacuum) that is described by the metric tensor gμν(x). In special theory of relativity, one uses x=(xμ)=(x0,x) with x0=ct,x=(−x1,−x2,−x3)=(−x,−y,−z) and signature (+,−,−,−). From the constancy of the speed of light in vacuum c in all inertial systems, S follows Einstein’s famous equation E = mc^2[21] and the invariance of the spacetime interval d2:=(x−y)2=(x0−y0)2+(x−y)2 under the so-called Lorentz transformations x→x′=Λx+a, where Λ is a real 4 × 4 matrix, and a is a four-dimensional translation vector. The laws of nature are invariant under the noncompact Poincaré group of transformations Λ,a. Today, physics is described by various fields, ϕ(x), and their symmetries.

To this end, the Lagrangian L=L(ϕ,∂μϕ) (this formulation also holds for GR where the field is given by the metric tensor gμν) is used to characterise the phenomena of nature, where the variables of the Lagrangian are the field ϕ and its first derivative ∂μϕ (to be considered as two different calculus variables x and y). The optimisation principle employed by nature is expressed by setting the variation δS=0 with the action given by S=∫Ldt=∫ℒd4x. This approach leads to the corresponding equations of motion, named the Euler equations. There is, however, a substantial variety of fields ϕ, which may be real or complex (charged) scalar fields (or pseudoscalar with respect to the Lorentz transformation Λ) of spin 0, which satisfy the Schrödinger (obtained from replacing observables by operators, i.e. E→iℏ∂∂t and p→−iℏ∇) or Klein–Gordon equations, or the scalar field of the Higgs boson. The well-known electrodynamics potential A^μ represents the photon of spin 1, whereas the metric tensor gμν, assumed to describe the hypothetical graviton, is of spin 2. The particles of matter, fermions, are represented by the Dirac (Weyl) four-spinor ψ^μ, and possess half-integer spin (electron, etc. 1/2, 3/2).^[22]

This means that the physics is represented by a collection of fields ϕ=ϕ(x) that are parametrised by points in four-dimensional spacetime x. However, there is a subtle difference among the fields, because the spacetime manifold, described by the collection of points x, is used to parametrise all other fields. In Part I [2], it was shown that there is strong evidence from the ESA Integral satellite measurements that spacetime may become discrete at a length scale of about 10⁻⁵⁰ m only, many orders of magnitude smaller than even the Planck length. The mass scale associated with the heaviest known particle, the top quark, ends at some 177 GeV/c² which corresponds to a length of about 10⁻¹⁸ m. In other words, material phenomena only see a spacetime continuum.

The most successful (and fully confirmed) theory of physics, the SM of particle physics, is characterised by the group SUc(3)⊗SUw(2)⊗U(1)em, where the indices stand for the strong (colour), weak, and electromagnetic interactions, respectively. These groups possess eight, three, and one generators, and hence the number of mediator bosons (exchange particles of physical forces), require that there exist eight gluons g (and eight antigluons g¯) for the strong force, three vector bosons (W±,Z0) for the weak interaction, and the photon γ for the electromagnetic force.

This theory is incomplete because gravity is not accounted for, and it also requires a set of 26 free parameters (if neutrinos with rest mass are to be included), which is not satisfactory. Furthermore, neither DM nor DE, nor the Higgs boson(s) are included in this group. In order to include the novel hypercomplex-gravity fields proposed by EHT the group structure needs to be extended further.

In contrast, SUSY exclusively describes gravity by a spin 2 boson, termed the graviton νGE, and other gravitational fields are unknown (except for the nonexisting but postulated gravitino of spin 3/2). Supersymmetry, first proposed in the 1970s, is an extension of the group underlying the SM and is one of the basic requirements of string theory, which aims at unifying gravity with the elementary particle forces. Supersymmetry tries to unify all material particles (i.e. fermions) with the mediators of the physical forces (bosons). This is not in accordance with the duality principle and thus – at least under EHT – is not a possible symmetry. In other words, the principle of duality does not permit SUSY. Therefore, the prediction of EHT is that supersymmetric partners cannot be found in our spacetime, and any attempt to produce these particles in even the most powerful accelerators will be in vain. An alternative must be sought.

The most widely known and highly developed theory beyond the SM is superstring theory, whose early roots date back to an accidental finding in 1968 by G. Veneziano concerning the beta function. Force carriers (bosons) and matter particles (fermions) are quantum fluctuations of the relativistic string. The most impressive feature of string theory is that it naturally predicts the existence of a spin 2 boson (assuming such a boson exists at all) as one of the quantum fluctuation modes of a heterotic string. Identifying this boson with the graviton, superstring theory unifies gravity with the three other elementary interactions. However, as was shown in the preceding sections, there is substantial experimental evidence holding against this concept. As will be shown below, there may exist extreme gravitomagnetic fields (hypercomplex-gravity fields) resulting from an interaction of electromagnetism and gravity (as predicted by EHT). Three additional gravitational particles (SU(2)) are required for these fields, namely, the spin 1 graviton ν~G, the gravitophoton ν~gp of spin 1, and the spin 0 quintessence particle ν~q (see the discussion on groups in Section 5.4).^[23] Hence, any further experiments (see Part I) reporting on the generation of these gravity-like fields (or hypercomplex-gravity fields) would be further experimental evidence in contradiction to string theory.

The novel idea chosen in EHT employs extra systems of numbers instead of extra spatial dimensions. This idea is a straightforward extension from the systems of numbers already used in physics, namely, the fields of real numbers ℝ and complex numbers ℂ. The next system of numbers are the hypercomplex numbers (or quaternions) ℍ, and thus in the next section, this field of numbers will be used in conjunction with orthogonal groups.

It should be noted that there is a homomorphism for hypercomplex groups O(n,q)=SO(2n) (a one-to-one mapping between hypercomplex groups and rotations), which results in a dynamical symmetry [61]. This means that when the original group O(n, q) is broken, its rank n is retained, independent of how the symmetry group is broken. Therefore, the number of Casimir operators C^m,m=1,…,n^[24] of the original group remains invariant regardless of the symmetry breaking. That is, the number of physical quantities (or quantum numbers characterising the physical system) that can be simultaneously measured does not change owing to symmetry breaking. As a result, the system will have n good quantum numbers (conserved physical quantities) that can be measured simultaneously together with the energy. These observables characterise the degeneracy of the energy eigenvalues of the system and determine the dimension of the multiplet (the total number of states that have almost the same energy eigenvalue or mass). For instance, group SU(3) has eight generators (Gell–Mann matrices λℓ,ℓ=1,…,8, where two of the generators are commuting. A multiplet comprising three quarks with total spin J = 1/2 and baryon number B = 1 (three quarks where two spins are antiparallel)^[25] characterises the energy of this state. Associated with this degenerated energy value is a multiplet of eight particles (octet) as depicted in the plane formed by the projection of the isospin number I₃ and the hypercharge Y, the Y−I3 plane, comprising the following particles: n, p, Σ−,Σ0,Σ+,Ξ−,Ξ+ and Λ, which is a singlet with isospin number I = 0. For J = 3/2 and B = 1 (three quark spins are parallel), where the baryon number B can be −1,0,+1, the multiplet is a decouplet. For the group SU(2), which is of rank 1, the number of multiplet states for a given angular momentum J is 2J + 1.

5.4 Cosmic Symmetry Group from Hypercomplex Numbers

If symmetry is the key to the cosmos, then this question naturally arises: What is the group structure of our cosmos? Is there a group describing the entire cosmos, i.e. both the stage (external spaces) and the actors (particles and fields)?

In EHT the cosmic group has to contain both the de Sitter spacetime dS1,3 and dual spacetime DdS1,3. In addition, all types of matter (fermions and bosons) as well as the associated internal gauge space (𝖧8) have to be accounted for. The cosmic group thus stands for a special symmetry between space and matter. The presentation that follows serves to demonstrate certain aspects of the nature of physical phenomena underlying the cosmos.

The cosmic group OC(32,ℍ), with q∈ℍ, where ℍ denotes the set of hypercomplex numbers (see the more detailed discussion in Section 5.5.3 of [9]), is considered the fundamental group for everything existing in the cosmos, i.e. both spacetime (stage) and matter (actors). However, the cosmic group should not be taken too literally as a group as it is immediately broken into a sufficient number of subgroups (32 = 2⁵ allows to build a group hierarchy with the sufficient number of subgroup levels) to arrive at physics supported by the experimental facts. The anticommutative algebra of hypercomplex numbers is employed as it seems to reflect the fundamental symmetry structure of nature. This includes (normal) gravitation in the form of Einsteinian gravitation, which may be only one aspect of the entire range of gravitational phenomena as well as the existence of extreme gravitomagnetic (or hypercomplex) fields.

Next, the symmetry of the cosmic group, OC(32,ℍ), has to be split into two groups (duality principle) in order to create the two separate entities of space and matter. Therefore, it is immediately obvious that matter cannot be solely derived from geometry. In the cosmic model of EHT, DE and the spacetime lattice are generated simultaneously owing to the duality principle (see above), and the total energy E_U = 0 (see Chapter 9 of [9]).

However, if the cosmic group is to represent the overall symmetry of the cosmos, a sequence of symmetry breaking processes is needed to produce the multitude of different physical phenomena observed in the cosmos. To this end, both spacetime and matter have to be initially created. Hence, symmetry needs to be broken further in order to obtain both the structure of matter, i.e. the families of particles, in conjunction with DE from which all material entities are derived and spacetime itself. Finally, a mechanism for the dynamics of the cosmos, that is, the expansion of spacetime (creation of motion), has to be provided.

This symmetry has to be broken if the cosmos is to enfold. To this end, the group OC(32,ℍ) is assumed to have been broken down into

where subscript S, denotes any type of space, i.e. both external and internal spaces, and M stands for all types of matter (fermions and bosons), i.e. OM and nonordinary matter (NOM). The cosmic group OC(32,ℍ) is thus separated (by spontaneous symmetry breaking) into two subgroups whose symmetries are at the roots of spacetime and matter. The cosmic group OC(32,ℍ) is the fundamental group for everything existing in the cosmos, i.e. both spacetime (in a very general sense) and all types of matter, including novel extreme gravitomagnetic fields or hypercomplex fields.

The cosmos comprises both external and internal space (duality principle). Therefore, for the description of the cosmic group, we start with the notion of the general space and its associated group OS(16,ℍ), which mathematically is assumed to describe all aspects of space in the universe.

Figure 5:

Spacetime as a geometric entity is considered to comprise (discrete) elemental surfaces or metrons (a term coined by the late B. Heim) that possess orientation; i.e. metrons are elemental surfaces with spin, either up or down, denoted by Δ_es where Δ marks the size of the metron surface. The spacetime grid comprised an extremely large number of metrons (definitely after inflation), forming practically a continuum as compared to the length scales involved in material processes and thus can be represented by a manifold; i.e. GR becomes the valid description. In other words, the spacetime lattice is a very rigid entity with regard to all energetic processes of matter. Nevertheless, this lattice is supposed to have started from a single metron. So long as the number of metrons is small, their interaction energy (potential energy) can be described by an Ising model, Est=−/21J(T)∑ijN(t)SiSj, where SiSj=+1 for parallel spins and SiSj=−1 for spins that are antiparallel. The energy is minimal if all spins are parallel. For large metron numbers, when the continuum picture holds, the Ising interaction energy goes over into a (smooth) Landau–Ginzburg potential, as is known from solid state theory.

The following symmetry breaking should not be taken literally, as the group OM(16,ℍ) will be used to describe the symmetry of the 15 hermetry families plus hermetry form 16 (see below). In order to proceed with an enfolding physical cosmos, the symmetry of the space group OS(16,ℍ) needs to be broken further in order to obtain the structure of matter (i.e. the families of particles), as well as spacetime itself, and to provide the means of an expanding spacetime.

Because of the duality principle (i.e. the existence of both external and internal space), the symmetry of this group is necessarily broken as

where group Oext(8,ℍ) describes the symmetries of the external space and OH(8,ℍ) is the above-cited Heim group (Section 4). The group Oext(8,ℍ) itself is broken into two groups denoted by

where

which represents a cosmos with spacetime, but without matter. In EHT spacetime is a lattice of elemental surfaces with spin. Its four generators stand for the two different atoms of spacetime (exo-spin and endo-spin atoms, see Fig. 5), which are distinguished by their orientation. The other two generators constitute the two particles of the semimaterial DE field, denoted by νde− (attractive to matter and repulsive for spacetime, positive energy density) and νde+ (repulsive to matter and attractive to spacetime, negative energy density) (for a more detailed discussion of the Ising model and DE, see Section 9.7.2) [9].

Dark energy is considered the precursor of matter, from which all material structures are ultimately derived. This group was broken (using set theory) into four subgroups O(1,ℍ), because in EHT only these four physical entities exist. At this stage of the cosmic evolution, none of the Higgs fields is present, i.e. group OH(8,ℍ) has not yet been activated; therefore, neither matter (in any form) nor inertia should exist.

From cosmological observations, it is known that inside a galaxy an acceleration a0=1.2×10−10 m/s² pointing towards the galactic centre must be obtained, which seems to be a global value for all galaxies. If DE is responsible for this acceleration, then the value of Λ inside a galaxy must change, and Einstein’s cosmological constant must be promoted to cosmological function Λ(x,t). Obviously, this can only be caused by the presence of the second field, the νde+ DE field (possessing negative energy density and contracting spacetime), whose interaction with visible matter inside the galaxy should lead to a repulsive gravitational force, substantially reducing the number of νde+ particles inside a galaxy. The only difference between intergalactic space and the space inside galaxies is the density of visible matter, whereby the density inside galaxies is about 10⁷ times larger. For the two different types of DE there are two corresponding cosmological constants that, in accordance with Einstein’s cosmological constant Λ, are denoted by Λ− (labelled negative for it causes spacetime to expand but possesses positive energy density) and Λ+ (labelled positive for it causes spacetime to contract but represents negative energy density). In general, there should be a complete symmetry between Λ+ and Λ−, that is, Einstein’s cosmological constant Λ should be zero – in a universe without matter. Because of the (small) presence of matter in intergalactic space, this symmetry is broken and Λ > 0 resulting in the current era. As a consequence of the presence of large amounts of visible matter inside galaxies (compared to intergalactic space), the value of Λ is supposed to change drastically.

In the present era, ∣Λ−∣>∣Λ+∣ (note that Λ−>0, Λ+<0) because an expansion of spacetime is observed. Note that the expansion or the arrow of time is similar to symmetry breaking as a preferred direction exists on the cosmic time scale and thus might be responsible for special physical phenomena. Therefore, Einstein’s cosmological constant, given by Λ=Λ−+Λ+, should be a function of time, and in this epoch, the value is Λ=Λ(t)>0. The value of Einstein’s cosmological parameter currently is Λ≃10−51 1/m², which is deemed to be responsible for spacetime expansion. It should be noted that for the pressure of the vacuum (intergalactic space) the equation

holds. That is, the current value of Λ exerts a negative pressure and thus causes an expansion of the spacetime lattice (universe). No singularities should exist (one of the fundamental principles of EHT); therefore, Λ(t) will have to change sign owing to the expansion process itself once the universe has reached a certain size. As the energy density of repulsive (regarding spacetime) DE is counted positive (Λ−,νde−), visible matter and negative DE attract each other. For attractive (regarding spacetime) DE (Λ+,νde+), the opposite effect occurs; that is, visible matter and positive DE repel each other. According to the EHT particle classification scheme, the boson mediating that force is termed quintessence particle, ν_q.

The two different particles of DE, each producing a substantial gravitational interaction by itself, Λ− (attractive to matter) and Λ+ (repulsive to matter), almost cancel each other in intergalactic space, and thus, the resulting Λ is extremely small. However, this does not mean that the two DE particles νde− and νde+→νde0 did combine into a single, only slightly repulsive DE particle. If this were the case, the MOND hypothesis could not be explained (see below). A further important consequence is that a postulated symmetry group SO(10) cannot be split into SO(8) ⊗ SO(2), whereas SO(2) is meant to describe DE. As DE is assumed to be described by two different particles (otherwise, there is no polarisation effect possible), this group cannot be correct as SO(2) has one generator only. In addition, the concept of a 10-dimensional space in which our four-dimensional spacetime is embedded is not needed in GR, because curvature is an intrinsic feature of spacetime. Moreover, the existence of a symmetry group SO(10) following from the concept of a 10-dimensional embedding space does not seem to be possible as such a group is no longer tenable owing to recent ACME results (see above). The Einstein value of Λ is valid in intergalactic space only where matter density is very low, and any polarisation effect owing to the presence of visible matter can be neglected, but this quasi-equilibrium is changed in the presence of visible matter, i.e. inside galaxies. Of course, the volume of all galaxies in the cosmos compared to the volume of the cosmos itself is negligible. Thus, Einstein’s cosmological constant Λ is valid on the cosmological scale, but not (locally) on the galactic scale.

It is postulated that Λ substantially changes inside galaxies because of gravitational polarisation. This weakens the Λ+ value and thus causes an acceleration towards the galactic centre. Moreover, Λ can also be changed by the local presence of extreme gravitomagnetic fields, and therefore, in general, Λ=Λ(x,t).

The effect of Λ+ (contraction of spacetime) being repulsive with regard to the visible matter inside a galaxy (because of its negative energy density) is largely neutralised inside a galaxy. Therefore, inside the galaxy, only the attractive gravitational effect of Λ− on OM remains. The reason why the Λ+ is neutralised inside a galaxy is due to the fact that a galaxy contains a large amount of OM, i.e. both visible matter inside the galaxy and DM (both visible matter inside the galaxy and DM in the halo with about 80 % of the matter in the halo). The surplus of Λ− cannot cause an expansion of spacetime because this is prevented by Newtonian gravitation of the visible mass inside the galaxy.

In order to provide an estimate for the magnitude of the MOND acceleration (here we consider only the observed measurements), we are resorting to several hand-waving arguments not meant to amount to a real derivation. The acceleration due to the DE field is given by the well-known equation

where r is the distance from the centre of the galaxy. It is assumed that this equation is valid up to the finite distance R_U. Assuming that the polarisation effect generates an opposite acceleration outside the galaxy (consider a simplified universe filled with DE particles, νde− and νde+, but which contains only a single galaxy), the MOND acceleration a₀ can be calculated as the acceleration averaged over the entire universe (the radius of the galaxy being negligible), and one obtains

The numerical value is given by

where the minus sign was chosen to indicate an acceleration towards the galactic centre, R_U denotes the Hubble radius, TU=13.82×109 years is the age of the universe, and Λ(TU)=1RU2, i.e. the amount of DE depends on the age of the universe.

There is presently a controversy about the validity of MOND. The research group of S. Mc Gaugh on November 13, 2018, published an article [64] in which the MOND hypothesis with a0=1.15×10−10 cm/s² has been confirmed contradicting the article by Rodrigues published on June 18, 2018, that arrived at the opposite result. In their reply to McGaugh’s article [65], [66], [67], they upheld their conclusions questioning the statistical approach by S. McGaugh et al.

This means that in the long run expansion needs to slow down and a minimal value has to be assumed. Perhaps at this point in time, quantum fluctuations can reverse the sign of Λ(TU), and contraction is initiated.

The coupling factor J in the Ising model (Fig. 5) generally depends on temperature T but may also depend on position (i.e. on indices i, j, k). The number of the atoms of space N(t) (sometimes also called spacetime atoms) depends on time (i.e. on the diameter D(t) of the universe). In the beginning phase of the evolution of the universe (as soon as the concept of temperature T could be employed, which is a macroscopic physical parameter), the interaction energy becomes a function of T, and as T = T(t), the interaction energy depends on the age of the universe itself. It is assumed that there is a critical temperature T_cr for which J(T)=0 for T<Tcr. At this particular cosmic time, t_tp, the expansion of the universe reaches its turning point, and the production rate of DE goes to 0 (contraction will set in). Although the value of the critical temperature T_cr is not known, it appears to be logical to identify temperature T with the temperature of the cosmic background radiation, currently at 2.7 K, which is regarded as some kind of universal cosmic temperature. If this symmetry breaking is comparable to a Bose–Einstein condensate, then at Tcr≈10−3 K symmetry breaking might set in, allowing us to speculate on the lifetime of the present universe. As was mentioned in the text, the grid spacing Δs (excluding pathological metrons that are extremely stretched; otherwise, one cannot use the concept of grid spacing) of the space lattice may be much smaller than the Planck length ℓ_Pl, and thus, the sum of discrete spins may be replaced by a continuous variable – a scalar bosonic field ϕ as will be discussed in more detail in Part III, see Section 9.7.2 in [9]. Hence, the application of Einstein’s field equations to a universe that only contains DE, being characterised by a cosmological parameter Λ=Λ(T(t)), and its geometry would be justified at practically all stages in the evolution of the universe, that is, after inflation.

The group OST(4,q), which is representing the actual stage(s) for physical events, is broken into

where one of the two OST(2,q) subgroups represent the dual spacetime DdS1,3 and the other one to group SO(1,3) (Minkowski spacetime dS1,3). In this case, the no-go Coleman–Mandula theorem (see M. Kaku [68]) does not apply, as the parameters of the Lie group O(2,q) are hypercomplex numbers that are anticommutative. Our (local) physical spacetime 𝖬4 (Minkowski space) is characterised by the symmetry group SO(1,3) of the Lorentz transformations (depending on the order of the coordinates, here x0=ct,x1=x,x2=y,x3=z is used). It is assumed that spaces dS1,3 and DdS1,3), (the latter of which is the location of negative DM) have the same spatial coordinates (spatial entanglement and differ only in the time coordinates ct and (ic′)(−it) with c′≈c due to spatial entanglement. Therefore, these two intertwined (entangled) spaces can be considered to comprise a five-dimensional de Sitter space dS2,3, because the product (ic′)(−it)>0. According to J. Malcadena [69], [70] (see also S. Carroll [71]), a five-dimensional anti-de Sitter space (AdS) (see A. Zee p. 662 [20]) with gravitation is equivalent to a flat four-dimensional QFT without gravitation. Everything that can happen physically in the five-dimensional AdS with gravitation is exactly analogous to a theory in the four-dimensional flat space without gravitation.

According to EHT, DM resides in dual space DdS1,3 and hence is principally unobservable in 𝖬4, i.e. only the gravitational effect of DM is active in our spacetime.

The symmetries of matter and physical interactions are governed by the Heim group. With respect to the Heim group, OH(8,q), one should remember that, as shown in (11), it is broken into the four subspaces

where for the moment only the symmetry group of the hermetry forms OH(3,q) is considered. The 15 generators of this group denote all possible particle families (not individual particles), whereas the individual particles are described by group OM(16,q), which consequently is broken into the two groups of fermions and bosons

where OB(8,q) and OF(8,q) denote the well-known symmetry groups for the bosons and fermions, respectively. In EHT, no supersymmetric particles exist. The physical meaning of these groups was already specified by (12) and (13). The two groups Oh+(2,q),Oh−(2,q) with its 12 generators denote the set of 12 charges or Higgs fields that are existing in physics; that is, there should be three Higgs bosons with masses m1=126 GeV/c² and m₂ and m₃, where m₂ and m₃ are supposed to be hypercomplex masses that are deemed to be responsible for the existence of extreme gravitomagnetic (or hypercomplex) fields that may have been measured by Tajmar and the proposed Heim experiment, respectively. In addition, there are organisational fields o₁, o₂ that are supposed to be the cause for the emergence of higher organisational structures. This is an expression describing the capabilities of nature to produce evolving organisational structures, enabling elementary particles to eventually form complex organic molecules. Furthermore, there are four quark charges q1,q2,q3,q4 that lead to a total of 15 gluons. Charges 11 and 12 are the two weak charges w1,w2 (or Isospin charges of fermions with values ±¹/₂). At the end of these three stages of symmetry breaking, four O(8,q) groups are obtained, denoted by

that are describing, respectively,

1. the stage(s) of the cosmos (the external spacetime Oext(8,q)

2. as well as the associated internal Heim space OH(8,q)

3. and the organisation of matter by the hermetry forms describing families of particles from which bosonic OB(8,q) and

4. fermionic OF(8,q) particles should eventually be obtained.

In the following, the meaning of the four O(8) subgroups will be explained further. Group Oext(8,q) contains all generators that stand for a cosmos without matter, which means both the atoms of spacetime and the two DE particles. Consequently, this group is further broken down

where the (primeval) subgroup Oc(4,q) (with four generators, two of them for the atoms of spacetime, that is, exospin and endospin atoms of space, and two generators for the two particles of the DE). The subgroup structure OST(4,q) contains the generators for a generalised spacetime encompassing our spacetime, dS1,3, (group SO(1,3)) and a dual de Sitter space DdS1,3 with imaginary speed of light ic and imaginary time −it. Space DdS1,3 is supposed to contain – the up to now in vain – the long-sought DM in the form of negative mass (for details, see Chapter 9.2 of [9]).

The Heim group OH(8,q) provides all existing particles and fields. The symmetry of the Heim group is broken (according to our set theory algorithm) into a 3, 2, 2, 1 subgroup structure

Here, we are interested in the group of hermetry forms OH(3,q) with its 15 generators that provide all families of particles. In the next step, the individual particles of these families and their mutual interactions are to be established (i.e. groups OB(8,q) and OF(8,q)). That is, the groups OB(8,q)⊗OF(8,q) are meant to describe all individual bosons and fermions.

In present physics, NOM does not exist. The eight hermetry forms of the outer cube comprise the DM neutrino ν_χ and the DM particle χ and the three gravitational bosons ν~G,ν~gp,ν~q from hypercomplex gravity, as well as the imaginary quark q_i, imaginary gluons g_I, and the imaginary photon γ_I. A hermetry form stands for a family of particles represented by its proper symmetry group. In this regard, there is no single supergroup structure in physics that contains all forces and particles. Instead, a hierarchy of groups seems to exist.

It should be noted that in EHT gravitation is characterised by a total of five gravitational constants GE,GN,Gp,Ggp,Gq and six gravitational particles: three for the cosmological gravitational fields (νGN,νgp,νq) and three for the extreme gravitomagnetic fields (ν~G,ν~gp,ν~q) that appear in the interaction between electromagnetism and gravity probably at cryogenic temperatures. Hence, these three particles are sometimes referred to as cold particles.

Figure 6:

The mandala of the physical forces according to EHT shows the four fundamental interactions, where the six gravitational bosons (three for the cosmological gravitational fields resulting from Einstein’s theory of general relativity, and the other three representing the postulated hypercomplex gravitational fields) are depicted in the upper half of the picture.

Newtonian gravitation, formulated in 1687, is supposed to be mediated by the graviton νGN, which is assumed to be a spin-2 tensor particle. Newton’s gravity was reformulated by Einstein in 1915 (GR). The coupling constant chosen by Einstein is G_N, but in EHT the value G_E is used that differs very slightly from G_N by the value of G_q in order to account for the interaction between matter and DE, causing spacetime to expand (in this respect, gravity can be considered repulsive). Therefore, gravity seems to have a multifaceted nature.

For the individual bosons, the group structure is given by the subgroup structure

where the group OB(8,q) comprises all existing physical field quanta. The subgroups SUGN(2) with three and SUG(2) with three generators are containing the three gravitational bosons of the cosmological gravitational fields (Newtonian gravity) and the three gravitational bosons representing the hypercomplex-gravity fields, resulting from the conversion of electromagnetic into gravitational fields as described above (Fig. 6).

The group OF(8,q) represents both the fermions of OM and NOM (negative and imaginary matter). The subgroup structure for the individual fermions is given by

where SUq(4) stands for the 15 quarks of EHT, index tq stands for top quark, and ν indicates the three neutrinos of the SM of particle physics. NOM is nonordinary matter, which goes beyond the SM, namely, the (virtual) electron of imaginary mass eI− and the quark of imaginary mass q_I. In addition, there is a fourth family of leptons that has negative mass, namely, the DM neutrino ν_χ of mass mνχ≈−3.2 eV and the DM particle χ of mass mχ≈−80 GeV. It should be noted that the masses of the forth family were already published in November 2015 [9], 1 year before the experimental results of the AMS spectrometer results were released, measured on board the International Space Station. The mass of the DM particle was finally published in October 2016 [72]. The explanation of the physical meaning of the other subgroups is discussed below.

For the definition of imaginary mass as used in [9] (cf. Section 5.5.2), electrons of imaginary mass, eI−, and quarks of imaginary mass, q_I, are assumed to result from some kind of delayed symmetry breaking. Instead of spontaneous symmetry breaking, the existence of these virtual particles of imaginary mass is postulated. They might have been generated, for instance, in the cryogenic rotating Nb ring in the experiments by Tajmar et al. at low temperature, and are not tachyons, because of their electromagnetic interaction. That is, particles of imaginary mass should not possess gravitational mass but should be subject to inertia. This should result in an interaction between electromagnetism and gravitation, which might explain the extraordinary strength of the possibly observed gravitomagnetic fields by Tajmar et al. The two DM particles, i.e. the fourth lepton υ_dm and the corresponding fourth neutrino ν_dm, are considered to have negative masses of −80.7 GeV/c² and −3.2 eV/c², respectively (see also Fig. 4). Neutrino oscillations are observed experimentally; that is, the various types of neutrinos can change into each other. However, DM particles are supposed to exist in dual spacetime DdS1,3, and therefore only their gravitational interaction with matter in our (de Sitter) spacetime dS1,3 can be observed, but not their speed. In other words, DM is dark, because the particles do not exist in our spacetime and hence cannot be directly measured.

After the Big Bang, the universe rapidly expanded from an incredibly small region with dimensions of 10⁻³³ cm and an unthinkably high energy density of 10⁹⁴ g/cm³ – this initial phase of the universe is known as the Planck aera. The Grand Unified Theories of today suggest…

W. Greiner [73]

At the time of the Big Bang,

the universe was concentrated at one point.

E. Zeidler [74], p. 227

7.1 Speeds of Light and Gravitation

Gravitation, identified with the Newtonian force formulated in 1687 and reformulated by A. Einstein in 1915 as the general theory of relativity, is still a mysterious force. Gravitation is supposed to be mediated by the graviton, νGN, a hypothetical spin-2 tensor particle according to modern quantum field theory. The force acting between two masses (m1,m2>0) is characterised by the gravitational coupling constant G_N (the index N stands for Newton), which is the same in both Newtonian and Einsteinian gravitation. Recent measurements of G_N (there is no theory to calculate its value) have shown strange deviations in the results despite the accuracy of the measurement techniques. So far, this problem remains unresolved [2]. It is generally assumed that the speed of photons in vacuum as obtained from Maxwell’s theory and the propagation speed of gravitational waves as utilised in GR are the same, namely, c=2.99792458×108 ms−1.

Calculations in Section 5.5.2 Non-Ordinary Matter in [9] (and possibly also experiments) suggest that gravity might have a more subtle structure. In EHT, gravity exhibits a multifaceted nature comprising three gravitational constants: G_p for hadrons, G_gp for leptons, and G_q for the interaction with DE (the vacuum field of spacetime) and the spacetime lattice (or, considered to be a continuum, depending on spatial resolution).

This means that the Newtonian gravitational constant should be a combination GN=Gp+Ggp. To account for the interaction of luminous matter with the vacuum field (DE, characterised by Einstein’s cosmological constant Λ), a further gravitational constant needs to be introduced. It is termed Einstein’s gravitational constant, because it plays a role only in GR, given by GE=GN+Gq. Now Newtonian and Einsteinian gravitations exhibit slightly different gravitational constants because in Newton’s theory space and time have absolute character (static), whereas in Einstein’s GR spacetime is a dynamical field generated in combination with the DE field (duality principle).

As a result, because of the presence of

owing to the existence of the DE field, a very small difference in the propagation speed of gravitational waves c_G and the speed of light in vacuum c could exist, given by the relation

However, when this relation was found, such a difference in propagation speed appeared to be far too small to be measured, notwithstanding the fact that at that time only indirect proof for the existence of gravitational waves existed. In 1974, Hulse and Taylor discovered a binary star system comprising a pulsar and a neutron star (a pulsar is a radiating neutron star, diameter about 10 km), termed PSR B1913+16 pulsar, orbiting their common centre of mass. The emanation of gravitational radiation was concluded by the rate of decrease of the orbital period at 76.5μ s per year in precise agreement with the loss of energy of 7.35×1024 W due to gravitational radiation as required by GR.

Nevertheless, according to EHT, gravitational waves should propagate slightly faster than electromagnetic waves. This effect is extremely small and can be seen only on Earth if there are detectors for gravitational wave signals and for measuring photons from a source several hundred million light-years away, that is, far outside the Milky Way galaxy. In addition, it must be possible to unambiguously associate gravitational waves and photons with such a source. Gravitational waves generated from colliding black holes or neutron stars should arrive somewhat earlier than the corresponding photons.

In his series of three lectures on string theory from January 29 to 31, 2007, at CERN, B. Zwiebach of Massachusetts Institute of Technology (MIT) predicted that the gravitational effects of strings by forming cusps and kinks propagating on string loops are expected to produce powerful bursts of gravitational waves to be observable by LIGO. Waves may even have values as small as GNμ≈10−13, where G_N is Newton’s constant, and μ denotes the mass per unit length and GN≈MPl−2 for ℏ=c=1. The LIGO, Wilkinson microwave anisotropy probe (WMAP), and COsmic Background Explorer (COBE) data have checked an enormous parameter space, especially the Advanced LIGO data from 2015–2016 for both stochastic and burst measurements, and comprehensive results were eventually published in May 2018 [77]. Not the slightest signal has been detected (cf. fig. 2 in [77]). It now seems to be justified to conclude that cosmic strings do not have physical reality. It is a sobering fact that measurable predictions issued by string theory over the years all have been invalidated by experimental data as soon as experimental techniques became sophisticated enough to overcome the substantial experimental thresholds posed by string theory predictions. Originally cosmic strings were proposed as the main source for cosmic density fluctuations, but this would require a value of GNμ≈10−5.5. This assumption was already ruled out by the COBE Satellite data. Density fluctuations are now thought to be the result of quantum fluctuations during the inflationary period. In later years, WMAP data reduced this value further below Gμ<10−7, lower than the value calculated from GUT, which is given by GNμ≈(M∗mPl)2=10−6, where M^∗ is the string mass, that, if it were to result from a GUT, would be M∗≈10−3MPl. Both the COBE and WMAP data delivered the unequivocal experimental message that GUT theories may not exist in physics.

Only a few years ago, such a scenario would have been impossible to measure, but with LIGO (the detection of gravitational waves by laser interferometry was conceived in the 1990s, but the first measurement took place in 2017), cosmology has entered the age of gravitational wave detection. In October 2017, double-messenger wave astronomy has been established [58] with the simultaneous observation of gravitational waves from a binary neutron star merger and the photons emitted by gamma ray burst GRB170817A. These observations led to a constraint of the speed of gravitational waves given by the inequality

which is almost the same numerical value as calculated from the above equation. This constraint also rules out most of the alternative tensor-scalar theories of GR. As was already shown on p. 363 in [9], the results of M. Kramer and N. Wex do not seem to leave any room for measurable deviations from Einstein’s GR.

The value for the speed of gravitational waves from EHT, c_G, is given by the equation

Because the gravitational waves and photon signals from the merging neutron stars arrived from a source 130 million light-years away, the path length difference of the two signals should be

where t_S denotes the signal travel time. As the measured distance in the arrival time for the two signals is Δt=1.7 s, one calculates

where t_S was converted into seconds. It should be noted that neither the expansion of the universe nor curvature was considered in the simple calculation for d_S. Compared to the measured value of 1.7 s, the agreement with the calculated time difference of 5.7 s appears reasonable, but it must be mentioned that according to EHT the gravitational wave should have arrived first. The effect of the DE field is very subtle, but definitely visible. It remains to be seen whether this interpretation can stand the test of time as there is no doubt that further binary systems of black holes and neutron stars will be found in the near future. The prediction is that there should be small differences in arrival times for gravitational waves and photons; i.e. the gravitational wave should always be slightly ahead of the photon signal. If this were the case, it would be a hint that novel physics may have been found.

7.2 Big Bang Scenario Questioned

As we have argued in [2], there is now substantial experimental evidence against the existence of real extra dimensions, questioning the validity of these Grand Unified Theories and their accompanying supersymmetric particles. As a consequence, the concept of a hot Big Bang should also be regarded critically. Cosmic inflation, that is, the exponential expansion of the spatial dimension of the initial universe by a scale factor of about 10⁴³ between the time period of 10⁻³⁶ s and 10⁻³⁴ s, is supported by the observed homogeneity of the CMB radiation. Otherwise, any two photons located at two opposite points as seen from Earth and separated by a distance larger than ¹/₂c T_U (T_U denotes the age of the universe) would not have had time to interact. Hence, without causal contact, the homogeneous distribution of the CMB radiation must be considered a purely random phenomenon – with practically zero probability to occur.

1. According to the fundamental principles laid out in Section 4.1, there are no singularities in physics. In other words, according to the Big Bang, the origin of the physical universe would be due to a nonphysical event.

2. Energy conservation would be maximally violated by a Big Bang, whereas there is no known physical process that does not respect the principle of the conservation of energy. While theoretically conceivable, this would run counter to all prevailing experience and experiment.

3. According to the arguments of R. Penrose [43], the probability for this to happen is virtually zero, pp. 726–732 and Chapter 1 of [78].

4. Moreover, the recent H.E.S.S. (High Energy Stereoscopic System) measurements [79], an array of ground-based Cherenkov telescopes, are clearly in contradiction to a Big Bang because none of the Big Bang predicted WIMPs could be detected. In physics, a theory is wrong if there exists a single experiment that is in conflict with it. Although the failure of the H.E.S.S. collaboration to see any WIMPs (as postulated by SUSY theories) should not (yet) be considered fully conclusive, it renders the Big Bang concept highly questionable and justifies the search for alternatives that are in better agreement with commonly accepted physical principles.

5. These measurements are independently corroborated by most recent data from the XENON Collaboration (November 2017) [80] obtained by the XENON1T experiment that has been and is searching for the nonrelativistic nonbaryonic component of the Λ cold DM model (Λ CDM) using underground detectors located deep inside the Gran Sasso massif. WIMPs should be detected from their nuclear recoils interacting with the 3200 kg of ultrapure He, which is an increase of two orders of magnitude with respect to the previous experiment. Since the February 2017 beginning of XENON1T, no WIMP events have been recorded, and WIMP masses above 10 GeV/c² seem to be excluded. The experiment continues with detectable mass limits being lowered and the collection of more data during winter 2018. As of June 2018, the CERN Atlas collaboration did not see the decay of any DE scalars to SM particles, nor were any DM particles detected, and no superpartners were found [81].

6. An even more stringent WIMP mass limit comes from the CERN CMS experiment that will be discussed in Section 7.4 together with the DM problem.

In order to have a universe that follows its own conservation principles, an alternate explanation to the Big Bang is suggested. By contrast, a universe that conserves energy at all times and is governed by the principles of duality, optimisation, and quantum fluctuation is supposed to have the following properties:

1. A universe with total energy

   (17)EU≡0,

   obeying global energy conservation not only from its very beginning but also asserting its validity at all times would be in accordance with the optimisation principle of nature as stated in Section 4 as well as Occam’s razor, a major guiding principle in physics.

2. The duality principle causes the spacetime lattice and DE to be generated simultaneously, where the structural energy of the spacetime grid, EST<0, is obtained from Szilárd’s energy principle (below) relating information and energy, whereas the scalar DE field has positive energy and is related to Einstein’s equivalence principle. Hence, the total energy of the universe is given by the equation

   (18)EU(t)=EST(t)+EDE(t)=0,

   where the positive and negative energies globally add up to zero at all times under the assumption that all forms of positive energy (e.g. DM, radiation, or baryonic matter) are ultimately converted from DE.

3. Dark energy is considered the source for all types of matter that exist in the universe. A Lagrangian L = 0 is not unusual because in SR ds=0 (spacetime interval) for photons. Any type of matter and radiation in our universe is supposed to originate from DE as a direct result of the principle of energy conservation.

4. Besides duality, the quantisation principle has proven to be a cornerstone of physics. In general, all physical entities are quantised, and this should also apply to spacetime itself (i.e. spacetime should form a lattice). However, the grid spacings of spacetime may be much smaller than the Planck length as indicated by the ESA Integral satellite measurements (Section 5.3). Hence, the universe is assumed to owe its existence to a Quantised Bang, which, in conjunction with duality, forms both the mutually dependent spacetime lattice and the DE field. According to EHT, the formation of the universe appears first in the form of the spacetime lattice (quickly turning into a continuum) and the scalar DE field. This is based on the two fundamental energy equivalence principles from Szilárd and Einstein, according to (2) and (3).

5. When quantum fluctuations created the very first elemental surface of spacetime, the very first quantum of DE was produced simultaneously such that the potential energy of the spacetime lattice and the ensuing DE field added up to zero – EU≡0 at all instants of time.

In a recent article, A. Ijjas et al. [75] challenge the scenario of an inflationary universe, claiming that new data from the Planck satellite have raised considerable doubt on the validity of inflation. The theory of inflation was introduced by A. Guth of MIT some 35 years ago. Since then, it has become a cornerstone of modern astrophysics and cosmology. The other main concept in current astrophysics is the Big Bang, in which all the energy in the universe allegedly emerged from some kind of spatial singularity. It is presently assumed that immediately after the Big Bang the early cosmos underwent inflation (exponential spatial expansion). The article by Ijjas et al. triggered a fairly harsh reaction from established science, eliciting a letter to the editors of Scientific American signed by 33 leading scientists, sharply rejecting the claim that Planck data have invalidated the empirical testing of inflation. The undersigned of this open letter state that about 14,000 articles by 9000 scientists have been published in support of inflation, and thus, there existed a major consensus for its correctness. The situation bears resemblance to the open letter from 90 years ago known as “100 authors against Einstein.” As Einstein simply remarked, “Why 100 authors?” If GR is wrong, one is enough, clearly pointing out that science is not democratic, but elite; sheer numbers do not count.^[27]

However, inflation (like GR and unlike the Big Bang concept) has passed several important empirical tests. Despite the fact that primordial gravitational ripples have not (yet) been found, inflation appears to be capable of solving the problem of flatness, homogeneity, and isotropy of the cosmos. A different question is whether inflation exclusively follows from the proposed Big Bang. It is obvious that the Big Bang is not in conformity with the quantisation principle that excludes physical singularities, even though singularities are mathematically admissible solutions of Einstein’s field equations. In the next section, an alternative picture for the birth of spacetime is presented. It employs the quantisation principle and also leads to an inflationary universe, but is based on the genesis of DE and the unfoldment of spacetime.

7.3 Origin of Dark Energy, Dark Matter, and Baryonic Matter

In 2011, Perlmutter, Riess, and Schmidt were awarded the Nobel Prize for the discovery of DE and the accompanying accelerated expansion of the universe.

In September 2017, claims were made [82], [83] that DE does not exist based on computer simulations by incorporating the observed (local) inhomogeneities in the distribution of baryonic matter and DM. The simplified cosmological model introduced by Einstein and Friedmann about 100 years ago is based on the assumption that, on average, the universe expands uniformly – as if all the matter were smoothly distributed in the cosmos. According to GR, observed inhomogeneities in curvature must act back on the expansion of the universe – at least in principle. This effect is termed backreacting and is supposed to be the cause for the expansion of the universe – according to the simulations. In addition, the inhomogeneous distribution of baryonic and DM may have another effect; namely, it may lead to a fractal dimension of spacetime because the radius of the universe may slightly vary, depending on the angular direction. Hence, this effect may produce small variations about the exact radius of the universe as calculated in the Einstein–Friedmann model of 1922. Further, as the universe is not exactly homogeneous, the matter distribution in the form of galaxies, clusters of galaxies, or voids may create slightly nonuniform expansion rates depending on the direction of observation.

Although it seems unlikely that backreacting is strong enough to explain the complete expansion of the universe, it might perhaps be sufficient to explain the accelerated expansion of the universe, because the expansion rate in any two directions should be different – owing to the local variations in matter density. In other words, the universe may expand almost uniform on the average (i.e. by calculating an expansion rate averaged over all angular directions), but no two directions will have exactly the same rate of expansion. Thus, might it be that the (claimed) accelerated expansion is an illusion, caused by the filament structure of matter, that is, it is an effect of the cosmic web itself?

7.3.1 Dark Energy, Inflation, and EHT

It should be kept in mind that without Einstein’s cosmological constant Λ (on the RHS of Einstein’s field equations) the CDT simulations of Ambjorn et al. do not result in a spacetime lattice. Further, their simulated spacetime is approximately four-dimensional, but appears to be fractal. Hence, all simulations that wish to do away with DE are not consistent with these results. Before any matter inhomogeneities can form, the spacetime lattice has to be in place, and this requires the Λ>0 value, needed to establish the arrow of time.

At a later instant of time in the cosmic evolution, when large-scale material structures in the universe had formed, they may indeed have led to a feedback effect on the curvature distribution in space and thus might be the cause for the small accelerated expansion. However, any independent simulations able to confirm this phenomenon must exhibit a realistic order of magnitude for the backreacting. Perhaps the universe is actually slowing down and does not accelerate beyond z = 0.7, caused by a possible backreacting effect.

In EHT, DE has to exist as it is generated simultaneously with the spacetime grid (Section 7.2). As an alternative to using the Ising model [9], once the spacetime lattice comprises a sufficiently large number of metrons (becomes a spacetime manifold), DE can be modelled by the cosmological field Λ(t)>0. This exerts a negative pressure in Einstein’s field equations causing the expansion of space, including inflation.

Recently, the following question from Nobel Laureate, astrophysicist, and cosmologist John Mather was posed: Can stars, by virtue of converting mass into energy, be responsible for the effects we attribute to DE (mentioned in the blog by E. Siegel on June 23, 2018) [84]. As is clear from the discussion in this section, under EHT, the DE field is the precursor of matter, and thus, no, it should be the other way around. Everything material in the universe ultimately is derived from DE.

The generation of matter is due to symmetry breaking that diminishes sharply the amount of DE. Thus, the inflation period comes to an end, driving the universe into slow expansion mode.

A speculative idea is that an overshoot might have taken place causing the universe to slow down too much, acting like a damped oscillator. Therefore, at a later period in the evolution, the universe may indeed be subject to a slightly accelerated expansion rate – as may be presently observed.

In 2016, the two expert groups for the neutron lifetime measurement (as discussed in Section 3.1.2 of Part I) [55] reanalyzed their data and confirmed that two different experimental techniques produced conflicting experimental findings. The first technique (termed bottle experiment) uses the storage of ultracold neutrons, counting the remaining neutrons and reporting a neutron lifetime of 878.5±0.8 s with a 68 % probability. The second experiment (termed beam experiment) detects the inelastically scattered neutrons, counting the protons resulting from neutron decay, measuring a lifetime of 887.7 ± 2.2 s. In EHT, owing to the postulated existence of hypercomplex masses, a free neutron can also decay according to the additional decay channel

n→pI++eI−+νe

that is much less likely than the well-known decay reaction

n→p++e−+ν¯e,

which means that the beam technique should actually report a larger neutron lifetime. Thus, this might be considered a hint for the existence of hypercomplex masses.

It is a firm belief in physics that all the known interactions in nature between material particles can be reduced to four interactions. However, Einstein’s gravity may not fall into this category because it is mediated by the curvature of spacetime and not necessarily by gravitational bosons. Of course, one can postulate the existence of these bosons,^[28] but they were never observed, nor are they necessary for gravity to function. GR is a geometric theory depending on external spacetime curvature, whereas the other interactions are living in internal or gauge space. Their interactions are not based on spacetime curvature, but instead by the exchange of bosons.

It was shown in Section 5 that in EHT there are two groups SUGE(2) and SUG(2) that formally describe three gravitational bosons νGE,νgp,νq that are mediating the forces for the cosmological fields as described by Einstein’s GR, whereas hypercomplex-gravity fields are described by the bosons ν~G,ν~gp,ν~q. However, there are fundamental physical differences between the two sets of gravitational generators that make us conclude that cosmological fields may not be working via mediator bosons but are of purely geometrical origin. Hence, it is more likely that the enormous weakness of Einstein’s gravity is the result of the enormous rigidity of the spacetime grid.

The following remarks on the nature of the two types of gravity seem to be in place.

Why is gravity so weak? The answer is twofold.

First, for the universe to become large, the gravitational constant G_E must be very small; otherwise, the curvature of spacetime would become too large. This also means that the grid spacing of the spacetime lattice (due to the quantisation principle) has to be discrete and extremely small.

Second, gravity may not be that small if the gravitational interaction between electromagnetism and gravity in the form of hypercomplex gravity is considered. However, this type of gravity is due to an exclusive interaction between material particles without involving the geometry of spacetime. Einstein’s GR rests on pure geometry; it is questionable if there are real bosons mediating this force. If they do exist, Einstein bosons would be of spin 2, while hypercomplex-gravity bosons are of spin 1 and the interaction strength is about 20 orders of magnitude stronger than for spin 2 bosons.

In EHT, there exist three different gravitational coupling constants for cosmological fields, denoted by Gp,Ggp,Gq. The gravitational constant of GR comprises all three parts, namely, G_p (gravitational constant for the proton and neutron or hadrons), Ggp (gravitational constant for leptonic particles carrying an electric charge), and G_q the gravitational constant that couples particles carrying energy to the DE field Λ=Λ(t,x)), because in later cosmological epochs Λ may have developed a slight dependence on position x, as the distribution of visible and DM may not be entirely homogeneous even when considering large cosmological scales (see Part I). That is,

GE:=Gp+Ggp+Gq,

where G_E is termed Einstein’s gravitational coupling constant, which is different from Newton’s coupling constant

GN:=Gp+Ggp,

that does not know anything about DE.

1. G_p is assumed to describe the gravitational coupling between two proton masses m_p, that is, Gpmpmp/ℏc.

Not all three gravitational interactions may be active in a process. If leptons (electrons) are present (e.g. atoms), the gravitational constant Ggp has to be added. For all practical purposes, Newton’s law remains unchanged, as for matter built from atoms the coupling is practically given by G_N, since Gq≈10−18GN [9]. The Newtonian gravitational constant is given by GN=Gp+Ggp for Newton’s theory of gravity is based on the concept of absolute time and space, that is, spacetime without DE. Hence, the gravitational interaction between matter and the DE field, characterised by G_q, cannot appear in Newtonian physics, and therefore, F=mGNa≠mia. Hence, Einstein’s equivalence principle is an approximate symmetry only, which is eventually broken due to the presence of DE. The strong variations in the measured values of G_N (or G_E) may be caused mainly by diurnal neutrino fluctuations affecting the values of the gravitational constants G_E and G_N by ±Ggp, depending on the Sun’s activity (11- to 12-year period) and/or the location of the laboratory (daily/nightly variation due to the rotation of the Earth with respect to the Sun).

Figure 7:

Based on Newton’s gravitational law, one would expect the rotational speed of stars around the centre of a galaxy to decrease with distance from the galactic centre. Based on the distribution of visible matter, it is clear (left figure) that this is not the case. Instead, for most galaxies, the rotational speed remains constant farther away from the galactic centre. This observation for the Coma cluster of galaxies (distance about 330 Mly) led Caltech astronomer Fritz Zwicky already in 1933 to conclude that dark matter (DM) (this term was coined by Zwicky) must exist. Today, we know the amount of DM to be about five times larger (in accordance with observed weak and strong gravitational lensing based on theory of general relativity) than visible matter, including plasma, dust, black holes neutron stars, and white and brown dwarfs. However, despite enormous experimental efforts during the last four decades, DM was never detected. Therefore, a modification of Newton’s gravitational law, the so-called modified Newtonian dynamics (MOND) hypothesis has been suggested by Milgrom in 1983 that correctly explains this deviation from Newton’s law. However, the right picture (surprisingly) shows the rotational speed of stars for a galaxy more than 10 billion lys away exactly following Newton’s law [85]. This clearly signifies the correctness of Newton’s law for the early universe, starkly questioning the MOND hypothesis, but at the same time demonstrates the absence of DM inside galaxies in the epoch of the early universe. On the other hand, a total absence of DM, because of the photon pressure in the young universe, would have eliminated all small-scale overdensities, thus reducing the matter density to the equilibrium value, while large-scale overdensities (for distances larger than the time needed by the speed of light to traverse them) should have scale invariant density fluctuations. With no DM present, the cosmic microwave background spectrum of the small-scale fluctuations should exhibit a dramatic reduction in its energetic amplitudes owing to radiation pressure. But this phenomenon has not been observed. Hence, the small-scale fluctuations must have been affected by an additional energy component in the universe (so-called silk damping) that is not subject to photon interaction (no radiation, no collisions). Computer simulations have proved that a universe without DM is not conceivable; i.e. the observed galactic cluster structures would not exist. Therefore, the currently observed distribution of DM should have evolved during the temporal evolution of the universe, but this leaves the DM enigma unresolved. Consequently, in Section 7.6, an attempt is presented to derive Milgrom’s formula without compromising Newton’s law. In addition, the postulated variation of the gravitational constant G_E as suggested by P. A. M. Dirac cannot have taken place. The Bullet cluster, which comprises two clusters of galaxies colliding at a high speed (about 10⁷ km/h), shows that its mass is distributed in two separated parts. When the optical image from gravitational lensing is overlaid with the inferred gravitational lensing picture of the DM matter distribution, two spatially separated locations are found (normally coloured pink and blue), providing clear evidence for the existence of DM.

7.4 Masses of Dark Matter Particles

The AMS-2 and PAMELA experiments [79], [95], [96], [97], [98], [99], [100], [101] have measured an energy resonance at 80 GeV, indicative of a particle of this energy that possibly might be considered to be the energy of the DM particle χ. However, the associated particle was not found.

However, as known from the latest LHC data as of winter 2018, there is no new particle of a mass around +80 GeV/c²; that is, there is no particle of positive mass in our spacetime as all accelerators (LEP, SPS, Fermilab, SLAC, SPS) have demonstrated for decades and, more recently, the LHC too. In particular, since the LHC upgrade in 2015 to an energy of s=13 TeV, an intensive search program for DM particles has been initiated by both the CMS and ATLAS collaborations. SLAC and Fermilab [102] initiated the Super CDMS (Cold Dark Matter Search) SNOLAB project, to start in the early 2020s, which is supposed to be 50 times more sensitive to very light DM particles than its predecessor, CDMS. Located at the Soudan Underground Laboratory, CDMS ended its operation in 2015 without any result, after decades of DM search. The new experiments sound more like an act of desperation, because very light particles never before were considered to be candidates of DM particles. We dare to predict that this last act in the drama of the search for DM particles will also leave the experimenters empty handed – DM particles are not living in our spacetime.^[30] Every LHC run has generated increasing experimental constraints on the mass of DM particles. The most recent results of the CMS collaboration [28] have not seen any hint for a DM particle down to a mass of 1 GeV and an associate mediator boson in the range of 50–100 GeV. There is no experimental evidence whatsoever for supersymmetric particles or technicolour or extra dimensions, and all of those theoretical constructs, based on these concepts, although being around for decades and having underwent numerous refinement processes, appear to be more and more unlikely as possible solutions to the hierarchy problem.

After the inflation period, the universe only comprised cold hydrogen gas, but was awash in background radiation, including radio waves. The cosmic afterglow may be attributed to the so-called Big Bang, but a different scenario may also be conceivable. In any case, the universe at about 380,000 years was dark. After several hundred million years, gravity, initiated by primordial density fluctuations, started to form massive stars. Their first light excited the hyperfine transition of the neutral hydrogen H₂ spectral line (the spin flip spectral line, s = ¹/₂ ℏ → −¹/₂ ℏ, for the hydrogen ground state n = 1) in the high redshift intergalactic medium, and caused it to absorb these radio waves at nominally 1420 MHz. Due to the expansion of the universe, the actual absorption frequency is lower, given by 1420/(1 + z) MHz, where z=λ−λ0λ0 and λ,λ0 denote the redshifted and laboratory (restframe)–measured wavelengths. This radiation should come from everywhere in space. J. D. Bowman, already in his Ph.D. thesis [103], suggested to measure these intrinsically faint signals masked by foreground emission and after 12 years deployed an antenna in Australia. This location was selected because of the low-level galactic noise radiation. There were first hints of a signal in early 2016 at a frequency of 78 MHz and depth of 0.5 K, with a frequency width of about 19 MHz. But the signal was twice as strong as theoretically predicted. The conclusion from this signal is that the absorbing hydrogen gas must have been colder as assumed. The conjecture is that DM is responsible for this phenomenon. R. Barkana [104] believes that hydrogen and DM particles are scattering each other. This should have appeared about 180 million years after the Big Bang. The substance of DM remains unknown. Signals from the first stars are predicted to arrive in the form of radio waves as recorded by the EDGES antenna of J. Bowman. Explanation from R. Barkana [105], [104]: There existed two actors in the form of first stars and DM. The radio waves were sent by the first stars, whereas the DM collided with the gas and cooled it down. Extra cold material explains the stronger than expected signal. Gravity by Einstein is correct. The conclusion by Barkana [105] is that DM therefore comprised particles. The discovery indicates that these should be low-mass particles, possibly several proton masses. However, no DM particle was ever seen, and these measurements are not proving that DM particles were encountered, but they may show the DM gravitational interaction. In other words, if the DM particles are in the dual space, and their gravitational potential is felt by the hydrogen gas, then a cooling effect by this potential is encountered. However, these signals are not proof for the existence of DM particles.

Numerous types of DM particles have been proposed over the years from the extension of the SM of particle physics. WIMPs are new but stable elementary particles with masses and coupling strengths at the electroweak scale, produced by the thermal background of the universe through an assumed hot Big Bang. However, in order to obtain today’s observed abundance of DM from this hypothesised thermal production, WIMPs need to self-annihilate in high DM density regions by a self-annihilation cross section that corresponds to a particle energy in the range between 80 and 100 GeV. Potential continuum emission of very high energy in the final state can be detected by the H.E.S.S. array of ground-based Cherenkov telescopes. WIMPs of higher mass were never detected in any of the accelerators, including the LHC of up to a particle energy of 1.7 TeV. These data are in accordance with a recent but totally different type of observation coming from the H.E.S.S. array of ground-based Cherenkov telescopes [79]. WIMP–DM particle DM annihilation signals focus on regions in the sky with both expected high DM density and reduced astrophysical γ-ray signal measure. Therefore, any very high energy γ-ray observations of the galactic centre (GC) region are among the most promising avenues to look for DM annihilation signals due to the GC proximity and its expected large DM content.

7.5 Dark Matter Space

It is a sobering fact that despite enormous, long-standing experimental efforts only indirect signs for the existence of DM have been found in the form of gravitational interaction with matter and radiation. So far, DM particles were never detected in our spacetime. On the other hand, a comprehensive mapping of the distribution of DM in the cosmos exists, based on its cosmic scale gravitational interaction.

The solution to this conundrum may be twofold. First, it needs to be accepted that there are no DM particles of positive mass m > 0. Hence, DM must have negative mass. However, this kind of exotic mass cannot exist in our spacetime. Second, as DM particles, denoted by χ, are known to exist but cannot be found in our spacetime, assumed to be a de Sitter space dS(1,3), they have to reside in a dual spacetime, termed DdS(1,3) [9], whose physical properties are discussed below. Such a dual space must allow the gravitational interaction of DM with OM of our spacetime, in order to be consistent with astrophysical observations. Furthermore, this gravitational interaction must be attractive. As no DM particle of positive mass was measured, the existence of a dual spacetime containing particles of negative mass m < 0 is postulated.

Next, the coordinate structure of this dual spacetime needs to be determined. The experimental proof of Newton’s law has been exhaustively validated at all scales. Therefore, i.e. no spatial coordinates x4,x5,… are available for an extension of our spacetime, and no Kaluza–Klein tower for the mass of the DM particle can be constructed. The only remaining option is time coordinate t. In particle physics, t > 0 is used for particles moving forward in time, while antiparticles are characterised by pμ→−pμ, i.e. the reversal of both time and space, together with charge conjugation. This means an antiparticle has its charge reversed and is moving backward in time t < 0 and space −x. Therefore, the only remaining coordinate extension is imaginary time. Hence, the coordinates of xμ∈DdS1,3 are assumed to be given by

where the speed of light must be chosen as ic in order to produce a positive particle energy. As will be shown below, no waves (radiation) should be possible in this space,

to guarantee the observed attractive gravitational force in dS1,3, resulting from mχ<0 of DM particles living in dual spacetime. The metric of the dual spacetime DdS1,3 is the same as for our spacetime dS1,3, but the physics is different. It should be noted that DM particles are of negative mass in dual space, but in our spacetime, only their positive energy is present – not their mass. It would be incorrect to think of a negative DM mass existing in our spacetime. In particular, no repulsive force between matter and DM can be constructed, for instance, by applying Newton’s law. This only works if both masses are present in our spacetime. Thus, the energy of DM particles, as observed in our spacetime, is positive, and the force exerted by the gravitational field of DM particles is attractive.

The question arises how the spatial coordinates of dual space are related to the spatial dimensions of our spacetime, because the gravitational interaction between the two spaces is a given experimental fact. The answer is straightforward; spaces dS1,3 and DdS1,3 share the spatial coordinates x, y, z, but time is real for dS1,3 and imaginary for DdS1,3. The transformation t→−it changes the character of the wave equation (Schrödinger) into a heat-type equation. Therefore, electromagnetic waves cannot exist in DdS1,3.

It remains to be determined what the different types of DM particles are. To this end, the discussion of the group OH(3,ℍ) (Section 5.4) should be recalled. Two additional particles (with respect to the SM) were proposed to exist, namely, the DM particle χ and DM neutrino ν_χ, constituting the fourth lepton family (Fig. 4).

Their masses, however, are negative, viz.

respectively, because they are not living in our spacetime; this is owing to dual spacetime DdS1,3, also termed dark spacetime. In order that the gravitational interaction of DM particles can be observed in our spacetime, dS1,3, the sharing of the spatial coordinates x1,x2,x3 is required. However, the respective energies associated with DM particles are positive,

as actually observed in our spacetime, owing to the physical properties of dark spacetime, that is, x0=ct with c as the speed of light in vacuum, expressed by icDdS1,3 so that the energy of the χ and ν_χ particles is counted positive in accordance with Einstein’s equation E=m(ic)2 as m < 0 giving E > 0. Hence, there should be no contradiction that only three particle families exist, because this requirement is valid only for our spacetime dS1,3 and for m > 0. Dark matter particles, on the other hand, are in dual space DdS1,3 with m < 0. Therefore, only the gravitational interaction of the DM particles with matter and radiation is perceived in our spacetime. The two DM particles themselves, in principle, cannot be measured in our spacetime. Consequently, no experiment should ever be able to directly see a DM particle. However, at present, there is no exact knowledge about the decay channel of DM particles.

There is a major physical difference between the physical properties of spacetimes dS1,3 and DdS1,3. Consider the process of generating light by atoms or molecules. The time-independent Schrödinger equation describes particles, e.g. atoms, with electrons having definite discrete energies Enℓ (ℓ (n,ℓ∈ℕ), where n is called the principal quantum number, and ℓ denotes the orbital angular momentum number. Electrons going from a higher (excited state) to a lower energy level are emitting radiation (light, photons); for instance, the visible spectrum of hydrogen has the two lines Hα and Hβ with frequencies ν of 4.47×1014 Hz and 6.17×1014 Hz and associated energy E=hν. Hence, the name luminous matter correctly describes that, and owing to this process, our universe is not dark. It is worthwhile to investigate the effect of the transformation from spacetime (luminous matter) to dual spacetime (supposed to contain dark, i.e. nonluminous matter), dS^1,3 → DdS^1,3, which is given by t→−it (spatial coordinates x remain unchanged). Consequently, the Schrödinger equation of dS^1,3 is transformed into a diffusion equation in DdS^1,3, that is,

where D is the diffusion coefficient. Note the all important sign changes on the RHS of the diffusion equation. In dual spacetime, there is no longer radiation (light); instead, DM seems to be governed by a diffusion equation that is not hyperbolic (wave propagation) but parabolic (heat conduction). In other words, there are no waves in dual spacetime.

Hence, dual spacetime should indeed be without luminosity. According to the diffusion equation, DM should be equally distributed in space 𝕊³, because the DM concentration always flows from regions of high density towards low density. This kind of equidistribution would be attained if there were no matter in our spacetime. Therefore, because of mutual gravitational interaction, DM will start to follow the lumps of OM in dS^1,3 deviating in the course of time from its initial primordial smooth distribution. Depending on the magnitude of the diffusion coefficient and the concentration of OM, this process may take several hundred million years. As indicated in Figure 7, the right picture is showing the rotation curve of stars in a young galaxy, when the universe was about several hundred million years old. As is clearly visible, the rotation curve is perfectly well described by Newton’s law. Therefore, it can be concluded that Milgrom’s hypothesis that Newton’s law does not describe the rotation curves of stars in galaxies is definitely incorrect – unless one assumes that the fundamental laws of physics are changing with time. There are no experiments whatsoever that are pointing into this direction. By contrast, some 10 billion years later, as indicated in the left picture, star rotation curves in spiral galaxies exhibit a substantial deviation from Newton’s law. Something must have happened. What could have changed is both the distribution of DE and DM inside and around galaxies. One might argue that the density of DE is far too small to have any gravitational influence inside galaxies. This argument would be correct, unless one assumes that two types of DE particles exist, generated in pairs mainly during the inflationary period, which are both attractive and repulsive. Due to the cosmic motion, there exists an asymmetry that slightly favours the production of repulsive DE particles, sustaining the cosmic expansion. The two types of DE particles must not be seen as particle and antiparticle that would annihilate each other. Dark energy is a precursor of matter and thus only exerts gravitational forces. However, the distribution of the two types of DE particles may be modified owing to the 10⁷ times higher mass density r inside galaxies, compared to the mass density of intergalactic space. This could lead to some kind of polarisation, that is, to a spatial separation of the DE particles. This topic will be discussed in more detail in Part III.

According to Bidin et al., DM is absent inside galaxies; therefore, an explanation for MOND cannot be based on the DM concept. Dark matter obviously was also not present in young galaxies, hence the modified rotation curves for the Milky Way and other galaxies (the graphs of orbital speed versus distance for other spiral galaxies are similar to our galaxy). On the other hand, it is known beyond doubt that DM has formed a DM halo around our Milky Way galaxy with a radius that may be 10 times as large as the halo of luminous matter. This can be interpreted as a clear sign that DM is attracted by ordinary mass. The DM halo is instrumental for the structural stability of galaxies.

At present, the relationship between spacetime and dark spacetime is not well understood.^[31] In particular, it is not known whether energy exchange beyond gravitational interaction can take place between the two spacetimes (remember the space is the same). For instance, as might have been measured by the AMS-02 experiment, a resonance may have been seen that shows increased proton-antiproton pair production, which could be interpreted as a hint for the decay of the DM particle by pair production at an energy of about 80 GeV. On the other hand, a particle with a mass in the range of 80 GeV/c² was ruled out from existing accelerator data a long time ago. This irreconcilable contradiction is resolved if DM particles of negative mass exist in DdS1,3, but this mass, because of its positive energy, is gravitationally attractive in dS1,3. However, it is not known if dark spacetime could act as a source of energy. In that case, the reverse process might also exist; i.e. there might be regions in our spacetime that appear void of matter and energy because of a hypothetical reverse reaction,

transferring energy back into dark spacetime. If this were the case, the normal annihilation process of particle and antiparticle should be suppressed, at least to a certain extent. However, such a process was not found in any accelerator data, and it needs to be understood under which circumstances energy transfer back into dark spacetime on a cosmic scale might be possible.

No claim can be made to have solved the DM riddle, but perhaps the novel concepts of four lepton families and dark spacetime might help to form the basis for a future solution.

7.6 Spacetime Lattice and Propagation Speeds

In this section, the possible impact of the lattice or grid structure of spacetime on energy levels and the propagation speed of matter as well as photons will be qualitatively discussed utilising a crude model adapted from solid state physics. Space is considered to be one-dimensional with the topology of a circle, but this is not important for the discussion.

Figure 8:

The left part of the figure shows both the temporal and spatial of evolution of the spacetime lattice of the universe, as described by an FLRW metric ds2=c2dt2−a2(t)(dr21−kr2+r2(dθ2+sin2dϕ2)), where a(t) denotes the cosmic scale factor (expansion factor) for the comoving coordinate system.