Topological-Geometrical and Physical Interpretation of the Dark Energy of the Cosmos as a “Halo” Energy of the Schrödinger Quantum Wave

The paper concludes that the energy given by Einstein’s famous formula E = mc2 consists of two parts. The first part is the positive energy of the quantum particle modeled by the topology of the zero set. The second part is the absolute value of the negative energy of the quantum Schr?dinger wave modeled by the topology of the empty set. We reason that the latter is nothing else but the so called missing dark energy of the universe which accounts for 94.45% of the total energy, in full agreement with the WMAP and Supernova cosmic measurement which was awarded the 2011 Nobel Prize in Physics. The dark energy of the quantum wave cannot be detected in the normal way because measurement collapses the quantum wave.

Nash Embedding of Witten’s M-Theory and the Hawking-Hartle Quantum Wave of Dark Energy

Euclidean embedding of the 11-dimensional M-theory turned out to require a very large space leaving lavish amounts of 242 dimensional pseudo truly empty “regions” devoid of space and time and consequently of anything resembling ordinary physical energy density. It is shown here using Nash embedding that the ratio of “solid” M-theory spacetime to its required embedding “non-spacetime” is 1/22 for a classical theory and 1/22.18033989 for an analogous fractal theory. This then leads to a maximal ordinary energy density equation equal to that of Einstein’s famous formula E=mc2 but multiplied with in full agreement with previous results obtained using relatively more conventional methods including running the electromagnetic fine structure constant in the exact solution of the hydrogen atom. Consequently, the new equation corresponds to a quantum relativity theory which unlike Einstein’s original equation gives quantitative predictions which agree perfectly with the cosmological measurements of WMAP and the analysis of certain supernova events. Never the less in our view dark energy also exists being the energy of the quantum wave amounting to 95.5 present of the total Einstein theoretical energy which is blind to any distinction between ordinary energy of the quantum particle and the dark energy of the quantum wave. However, since measurement leads to the collapse of the Hawking-Hartle quantum wave, dark energy being a quantum wave non-ordinary energy could not possibly be measured in the usual way unless highly refined quantum wave non-demolition technology is developed if possible. It is a further reason that dark energy having a different sign to ordinary energy is the cause behind the anti gravity force which is pushing the universe apart and accelerating cosmic expansion. Consequently it can be seen as the result of anticlastic Cartan-like curvature caused by extra compactified dimensions of spacetime. A simple toy model demonstration of the effect of curvature in a “material” space is briefly discussed.

If Quantum “Wave” of the Universe Then Quantum “Particle” of the Universe: A Resolution of the Dark Energy Question and the Black Hole Information Paradox

We start from a minimal number of generally accepted premises, in particular Hartle-Hawking quantum wave of the universe and von Neumann-Connes’ pointless and self referential spacetime geometry. We then proceed from there to show, using Dvoretzky’s theorem of measure concentration, that the total energy of the universe is divided into two parts, an ordinary energy very small part which we can measure while most of the energy is concentrated as the second part at the boundary of the holographic boundary which we cannot measure in a direct way. Finally the results are shown to imply a resolution of the black hole information paradox without violating the fundamental laws of physics. In this way the main thrust of the two opposing arguments and views, namely that of Hawking on the one side and Susskind as well as tHooft on the other side, is brought to a consistent and compatible coherent unit.

Quantum Entanglement: Where Dark Energy and Negative Gravity plus Accelerated Expansion of the Universe Comes from

Dark energy is shown to be the absolute value of the negative kinetic energy of the halo-like quantum wave modeled mathematically by the empty set in a five dimensional Kaluza-Klein (K-K) spacetime. Ordinary or position energy of the particle on the other hand is the dual of dark energy and is contained in the dynamic of the quantum particle modeled by the zero set in the same five dimensional K-K spacetime. The sum of both dark energy of the wave and the ordinary energy of the particle is exactly equal to the energy given by the well known formula of Einstein E=mc2 which is set in a four dimensional spacetime. Various interpretations of the results are presented and discussed based on the three fundamental energy density equations developed. In particular where E is the energy, m is the mass and c is the speed of light, is Hardy’s quantum entanglement and gives results in complete agreement with the cosmological measurements of WMAP and Supernova. On the other hand gives an intuitive explanation of negative gravity and the observed increased rate of cosmic expansion. Adding E (ordinary) to E (dark) one finds which as we mentioned above is Einstein’s famous relativity formula. We conclude that similar to the fact that the quantum wave interpreted generally as probability wave which is devoid of ordinary energy decides upon the location of a quantum particle, it also exerts a negative gravity effect on the cosmic scale of our clopen, i.e. closed and open universe. Analysis and conclusions are framed in a reader friendly manner in Figures 1-14 with detailed commentary.

What Is the Missing Dark Energy in a Nutshell and the Hawking-Hartle Quantum Wave Collapse

We reason that in quantum cosmology there are two kinds of energy. The first is the ordinary energy of the quantum particle which we can measure. The second is the dark energy of the quantum wave by quantum duality. Because measurement collapses the Hawking-Hartle quantum wave of the cosmos, dark energy cannot be detected or measured in any conventional manner. The quantitative results are confirmed using some exact solutions for the hydrogen atom. In particular the ordinary energy of the quantum particle is given by E(0) = (/2)(mc2) where is Hardy’s probability of quantum entanglement, =( - 1)/2 is the Hausdorff dimension of the zero measure thin Cantor set modeling the quantum particle, while the dark energy of the quantum wave is given by E(D) = (5/2)(mc2) where is the Hausdorff dimension of the positive measure thick empty Cantor set modeling the quantum wave and the factor five (5) is the Kaluza-Klein spacetime dimension to which the measure zero thin Cantor set D(0) = (0,) and the thick empty set D(-1) = (1,) must be lifted to give the five dimensional analogue sets namely and 5 needed for calculating the energy density E(0) and E(D) which together add to Einstein’s maximal total energy density E(total) = E(0) + E(D) = mc2 = E(Einstein). These results seem to be in complete agreement with the WMAP, supernova and recent Planck cosmic measurement as well as the 2005 quantum gravity experiments of V. V. Nesvizhersky and his associates. It also confirms the equivalence of wormhole solutions of Einstein’s equations and quantum entanglement by scaling the Planck scale.

A Quantum Space Model of Cosmic Evolution: Dark Energy and the Cyclic Universe

We present a Quantum Space Model (QSM) of cosmic evolution based on the theory that space consists of energy quanta from which our universe came about. We used the Friedmann equations to trace its history and predict its ultimate fate. Results provide further support to our recent proposal that the accelerating expansion of the universe is due to a scalar space field which has become known as Dark Energy. In our model, the universe started from high energy space quanta which were triggered by quantum fluctuations that caused the Big Bang. It then expanded and cooled undergoing phase transitions to radiation, fundamental particles, and matter. Matter agglomerated and grew into stars, galaxies, etc. and was eventually consolidated by gravity into Black Holes, which finally ended in a Big Crunch in a state of deep freeze inside the Black hole at 1.380 trillion years. Fluctuations, quantum tunneling, or some other mechanisms caused a new Bang to start another cycle in its life. Our results are in good agreement with the theoretical predictions of a cyclic universe by Steinhardt and his associates, and by Penrose. Space and energy are equivalent as embodied in the Planck energy equation. They give rise to the two principal long range forces in the universe: the gravitational force and the space force. The latter may be the fifth force in the universe. The two forces could provide the clockwork mechanism operating our cyclic universe. If the Law of Conservation of Energy is universal, then the cosmos is eternal.

A Topological Magueijo-Smolin Varying Speed of Light Theory, the Accelerated Cosmic Expansion and the Dark Energy of Pure Gravity

The paper presents a detailed analysis of ordinary and dark energy density of the cosmos based on two different but complimentary theories. First, and starting from the concept of the speed of light being an average over multi-fractals, we use Magueijo-Smolin’s ingenious revision of Einstein’s special relativity famous formula E = mc2 to a doubly special formula which includes the Planck energy as invariant to derive the ordinary energy density E(O) = mc2/22 and the dark energy density E(D) = mc2(21/22) wheremis the mass andcis the speed of light. Second we use the topological theory of pure gravity to reach the same result thus confirming the correctness of the theory of varying speed of light as well as the COBE, WMAP and Type 1a supernova cosmological measurements.

The hyperbolic Extension of Sigalotti-Hendi-Sharifzadeh’s Golden Triangle of Special Theory of Relativity and the Nature of Dark Energy

Previous work by Sigalotti in 2006 and recently by Hendi and Sharifzadeh in 2012 showed that all the fundamental equations of special relativity may be derived from a golden mean proportioned classical-Euclidean triangle and confirmed Einstein’s famous equation E=mc2. In the present work it is shown that exchanging the Euclidean triangle with a hyperbolic one an extended quantum relativity energy equation, namely , is obtained. The relevance of this result in understanding the true nature of the “missing” so-called dark energy of the cosmos is discussed in the light of the fact that the ratio of to E=mc2 is which agrees almost completely with the latest supernova and WMAP cosmological measurements. To put it succinctly what is really missing is a quantum mechanical factor equal 1/22 in Einstein’s purely relativistic equation. This factor on the other hand is derivable from the intrinsic hyperbolic Cantor set nature of quantum entanglement.

An Invitation to El Naschie’s Theory of Cantorian Space-Time and Dark Energy

The paper is a condensed but accurate account of El Naschie’s theory of Cantorian space-time which was used by him to clarify some major problems in theoretical physics and cosmology. In particular El Naschie’s revision and completion of relativity theory and demystification of dark energy are destined to be two milestones in the history of theoretical physics.

A Fractal Menger Sponge Space-Time Proposal to Reconcile Measurements and Theoretical Predictions of Cosmic Dark Energy

The 95.5 percent of discrepancy between theoretical prediction based on Einstein’s theory of relativity and the accurate cosmological measurement of WMAP and various supernova analyses is resolved classically using Newtonian mechanics in conjunction with a fractal Menger sponge space proposal. The new energy equation is thus based on the familiar kinetic energy of Newtonian mechanics scaled classically by a ratio relating our familiar three dimensional space homology to that of a Menger sponge. The remarkable final result is an energy equation identical to that of Einstein’s E=mc2 but divided by 22 so that our new equation reads as . Consequently the energy Lorentz-like reduction factor of percent is in astonishing agreement with cosmological measurements which put the hypothetical dark energy including dark matter at percent of the total theoretical value. In other words our analysis confirms the cosmological data putting the total value of measured ordinary matter and ordinary energy of the entire universe at 4.5 percent. Thus ordinary positive energy which can be measured using conventional methods is the energy of the quantum particle modeled by the Zero set in five dimensions. Dark energy on the other hand is the absolute value of the negative energy of the quantum Schrodinger wave modeled by the empty set also in five dimensions.

Dark Matter and the Energy-Momentum Relationship in a Hydrogen Atom

Einstein derived the energy-momentum relationship which holds in an isolated system in free space. However, this relationship is not applicable in the space inside a hydrogen atom where there is potential energy. Therefore, in 2011, the author derived an energy-momentum relationship applicable to the electron constituting a hydrogen atom. This paper derives that relationship in a simpler way using another method. From this relationship, it is possible to derive the formula for the energy levels of a hydrogen atom. The energy values obtained from this formula almost match the theoretical values of Bohr. However, the relationship derived by the author includes a state that cannot be predicted with Bohr’s theory. In the hydrogen atom, there is an energy level with n = 0. Also, there are energy levels where the relativistic energy of the electron becomes negative. An electron with this negative energy (mass) exists near the atomic nucleus (proton). The name “dark hydrogen atom” is given to matter formed from one electron with this negative mass and one proton with positive mass. Dark hydrogen atoms, dark hydrogen molecules, other types of dark atoms, and aggregates made up of dark molecules are plausible candidates for dark matter, the mysterious type of matter whose true nature is currently unknown.

NASA’s EM Drive Thrust from the Forces of the Quantum Vacuum of Spacetime

Building on the various manifestations of the forces latent in the quantum vacuum of spacetime such as Hawking’s radiation and Unruh temperature, we resolve a major paradox connected to an immensely important proposal by NASA scientists for constructing a practically fuelless spacecraft. In a nutshell, preliminary laboratory work shows that NASA’s electromagnetic drive project is viable and several experiments and measurements show it is real. Yet the proposal violates a fundamental principle of classical mechanics, namely Newton’s third law. The resolution of this paradox is quite straight forward in principle. It is simply the case that although the proposal seems to be based on classical mechanics and classical thinking it is only superficially so. Deep at the roots, the EM drive proposal of NASA is not classical physics but rather based on the vacuum forces of quantum cosmology and the theory of dark energy density of the universe. In fact the proposal is deeply linked to Hawking’s radiation and Unruh temperature, which is explained in some detail in the main body of the present short paper within the frame work of E-infinity Cantorian spacetime theory and D. Gross’ Heterotic superstring theory. In short the quintessence of our explanation is to regard the EM drive as a quasi electromagnetic cavity with an effective event horizon akin to that of a Hawking black hole emitting radiation causing ultimately the needed thrust to push the spacecraft forwards. In addition and by invoking fractal spacetime self similarity we show that a spacecraft will be subject to another cosmic thrust on the large scale of the entire cosmos.

The Electromagnetic Nature of Gravitation and Matter-Antimatter Antigravity. Surmise on Quantum Vacuum Gravitation and Cosmology

We show that the electromagnetic quantum vacuum derives directly from Maxwell’s theory and plays a primary role in quantum electrodynamics, particle physics, gravitation and cosmology. It corresponds to the electromagnetic field ground state at zero frequency, a zero-energy cosmic field permeating all of space and it is composed of real states, called kenons (κενο = vacuum). Photons are local oscillations of kenons guided by a non-local vector potential wave function with quantized amplitude. They propagate at the speed imposed by the vacuum electric permittivity ε0 and magnetic permeability μ0, which are intrinsic properties of the electromagnetic quantum vacuum. The electron-positron elementary charge derives naturally from the electromagnetic quantum vacuum and is related to the photon vector potential. We establish the masse-charge equivalence relation showing that the masses of all particles (leptons, mesons, baryons) and antiparticles are states of the elementary charges and their magnetic moments. The equivalence between Newton’s gravitational law and Coulomb’s electrostatic law results naturally. In addition, we show that the gravitational constant G is expressed explicitly through the electromagnetic quantum vacuum constants putting in evidence the electromagnetic nature of gravity. We draw that G is the same for matter and antimatter but gravitational forces should be repulsive between particles and antiparticles because their masses bear naturally opposite signs. The electromagnetic quantum vacuum appears to be the natural link between quantum electrodynamics, particle physics, gravitation and cosmology and constitutes a basic step towards a unified field theory. Dark Energy and Dark Matter might originate from the electromagnetic quantum vacuum fluctuations. The calculated electromagnetic vacuum energy density, related to the cosmological constant considered responsible for the cosmic acceleration, is in good agreement with the astrophysical observations. The cosmic acceleration may be due to both “quantum vacuum fluctuations” and “matter-antimatter gravitational repelling”. All the above results are established without stating any assumptions or postulates. Next, we advance two hypotheses with cosmological impact. The first is based on the possibility that gravitation is due to the electromagnetic quantum vacuum density of states fluctuations giving rise to a photon pressure at the characteristic collective oscillation frequencies of the charge densities composing the bodies (Electromagnetic Push Gravity). The second advances that energy, matter and antimatter in the universe emerge spontaneously from the quantum vacuum fluctuations as residues that remain stable in space and we present the main principles upon which a new cosmological model may be developed overcoming the well-known Big Bang issues.

Dark Energy from Kaluza-Klein Spacetime and Noether’s Theorem via Lagrangian Multiplier Method

The supposedly missing dark energy of the cosmos is found quantitatively in a direct analysis without involving ordinary energy. The analysis relies on five dimensional Kaluza-Klein spacetime and a Lagrangian constrained by an auxiliary condition. Employing the Lagrangian multiplier method, it is found that this multiplier is equal to the dark energy of the cosmos and is given by where E is energy, m is mass, c is the speed of light, and λ is the Lagrangian multiplier. The result is in full agreement with cosmic measurements which were awarded the 2011 Nobel Prize in Physics as well as with the interpretation that dark energy is the energy of the quantum wave while ordinary energy is the energy of the quantum particle. Consequently dark energy could not be found directly using our current measurement methods because measurement leads to wave collapse leaving only the quantum particle and its ordinary energy intact.

From a Dual Einstein-Kaluza Spacetime to ‘tHooft Renormalon and the Reality of Accelerated Cosmic Expansion

We use a dual Einstein-Kaluza spacetime to calculate the exact energy density of dark energy and dark matter using a novel topological computation method. Starting from the said spacetime and ‘tHooft’s topological renormalon as well as the corresponding symmetry group, we show how the zero set quantum particle and the empty set quantum wave interact with the vacuum and give rise to pure dark energy and pure dark matter all along with ordinary energy density of the cosmos. The consistency of the exact calculation and the accurate observations attests to the reality of ‘tHooft’s renormalon dark matter, pure dark energy and accelerated cosmic expansion.

On El Naschie’s Fractal-Cantorian Space-Time and Dark Energy—A Tutorial Review

This tutorial review is dedicated to the work of the outstanding Egyptian theoretical physicist and engineering scientist Prof. Mohamed El Naschie. Every physics student knows the well-known Einstein’s mass-energy equation, E=mc2, but unfortunately for physics, few know El Naschie’s modification, E(O)=mc2/22, and El Naschie’s dark energy equation E(D)=mc2(21/22) although this new insight has truly far reaching implications. This paper gives a short tutorial review of El Naschie’s fractal-Cantorian space-time as well as dark energy. Emphasis is put on the fundamental concept of Cantor set, fractal dimensions, zero set, empty set, and Casimir effect.

A Casimir-Dark Energy Nano Reactor Design—Phase One

A phase one design of a new free energy nano reactor is presented. The design is based on a basically topological interpretation of the Casimir effect as a natural intrinsic property of the geometrical topological structure of the quantum-Cantorian micro spacetime. In particular we view dark energy, Hawking negative energy, Unruh temperature and zero point vacuum energy as being different sides of the same multi-dimensional coin. This new interpretation compliments the earlier conventional interpretation as vacuum fluctuation or as a Schwinger source and links the Casimir energy to the so-called missing dark energy density of the cosmos. We start with a general outline of the theoretical principle and basic design concepts of a proposed Casimir dark energy nano reactor. In a nutshell the theory and consequently the actual design depend crucially upon the equivalence between the dark energy density of the cosmos and the faint local Casimir effect produced by two sides boundary condition quantum waves. This Casimir effect is then colossally amplified as a one internal quantum wave representing a Hartle-Hawking state vector of the universe pushing from the inside against the boundary of the universe with nothing balancing it from the non-existent outside. This strange situation becomes completely natural and logical when we remember that the boundary of the universe is a one sided M&#246bius like manifold. In view of the present theory, this is essentially what leads to the observed accelerated expansion of the cosmos. As in any reactor, the basic principle in the present design is to produce a gradient so that the excess energy on one side flows to the other side. Thus in principle we will restructure the local topology of space using material nanoscience technology to create an artificial local high dimensionality with a Dvoretzky theorem like 96 percent volume measure concentration. Without going into the intricate nonlinear dynamics and technological detail, it is fair to say that this would lead us to pure, clean, free energy obtained directly from the topology of spacetime via an artificial singularity. Needless to say, the entire design is based completely on the theory of quantum wave dark energy proposed by the present author for the first time in 2011 in a conference held in the Bibliotheca Alexandrina, Egypt and a little later in Shanghai, Republic of China. The quintessence of the present theory is easily explained as the Φ3 intrinsic Casimir topological energy where Φ=?(√5-1)/2 is produced from the zero set Φ of the quantum particle when we extract the empty set quantum wave Φ2 from it and find Φ-Φ2=Φ3 by restructuring space via conducting but uncharged plates similar to that of the classical Casimir experiments. Our proposed preliminary design of this Casimir-spacetime artificial singularity reactor follows in a natural way from the above.

Application of Dvoretzky’s Theorem of Measure Concentration in Physics and Cosmology

Using Dvoretzky’s theorem in conjunction with Bohm’s picture of a quantum particle inside a guiding quantum wave akin to De Broglie-Bohm pilot wave we derive Einstein’s famous formula E = mc2 as the sum of two parts E(O) = mc2/22 of the quantum particle and E(D) = m c2 (21/22) of the quantum wave where m is the mass, c is the speed of light and E is the energy. In addition we look at the problem of black holes information in the presence of extra dimensions where it seems initially that extra dimensions would logically lead to a hyper-surface for a black hole and consequently a reduction of the corresponding information density due to the dilution effect of these additional dimensions. The present paper argues that the counterintuitive opposite of the above is what should be expected. Again this surprising result is a consequence of the same well known theorem on measure concentration due to I. Dvoretzky. We conclude that there are only two real applications of the theorem and we expect that many more applications in physics and cosmology will be found in due course.

Between Quantum Mechanics and General Relativity

The origin of elementary particle mass is considered as a function of n-valued graviton quanta. To develop this concept we begin in a cold region of “empty space” comprised of only microscopic gravitons oscillating at angular frequency ω. From opposite directions enters a pair of stray protons. Upon colliding, heat and energy are released. Customarily, this phase and what follows afterward would be described by Quantum Chromodynamics (QCD). Instead, we argue for an intermediary step. One in which neighboring gravitons absorb discrete amounts of plane-wave energy. Captured by the graviton, the planewave becomes a standing wave, whereupon its electromagnetic energy densities are converted into gravitational quanta. Immediately thereafter an elementary particle is formed and emitted, having both mass and spin. From absorption to conversion to emission occurs in less than 3.7 × 10−16 s. During this basic unit of hybrid time, general relativity and quantum physics unite into a common set of physical laws. As additional stray protons collide the process continues. Over eons, vast regions of spacetime become populated with low-mass particles. These we recognize to be dark matter by its effects on large scale structures in the universe. Its counterpart, dark energy, arises when the conversion of gravitational quanta to particle emission is interrupted. This causes the gravitational quanta to be ejected. It is recognized by its large scale effects on the universe.

On the Real Einstein Beauty E = Kmc2

The paper suggests that E = mc2 may be open to misinterpretation and that in this form it is not what Einstein advanced first. It is further suggested to return to the slightly less compact formula E = Kmc2 where a < K < 1 which has the merit of accounting for the measured ordinary energy density of the cosmos (K = 1/22) and the conjectured missing dark energy density of the universe (K = 21/22) from the view point of economical notation.