properties of an expanding universe based on an ...

12/29/2017

Properties Of An Expanding Universe Based On An Expanding 2 D Singularity Folded Surface

PROPERTIES OF AN EXPANDING UNIVERSE BASED ON AN EXPANDING 2-D SINGULARITY FOLDED SURFACE Author and Original Researcher:

Mark S. Rogers, St. Petersburg, Florida

Key Words: Cosmology, Singularity, Event Horizon, Quantum Theory, Skeleton Geometry, Quantum Potential, Bra-Ket Notation, Quantum Energy Transport HYPOTHESIS The Universe could not have started from a 1R point source Singularity that expanded rapidly radially outward into a 3R to create time, space, matter, and energy including the vacuum of a 3D space, but we propose rather that a 1R point Singularity expanded into a 2R Singularity surface that folded and expanded rapidly into the plane possessing characteristics of folded geometry, infinite expansion, and infinite energy and which created the 3-D universe in 3R on the posterior side of the expanding event horizon. This work maintains consistency with the Copenhagen interpretation of Heisenberg’s quantum mechanics. ACKNOWLEDGEMENTS I am deeply indebted to Dr. Efstratios Manousakis for his seminal paper “Founding Quantum Theory on the Basis of Consciousness”1 and his approach which serves as a template for many of the arguments contained herein, and to J.C.S. Neves for his inspirational publications noted within the references, and for his unselfish support and kind comments during the evolution of this work.

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INTRODUCTION AND BACKGROUND: There can be no question that the study of Singularities23 within the context of General Relativity45, reveal issues that have historically been problematic in Big Bang6, black hole789 and cosmological models1011. Contemporary solutions to cosmological models12 reveal problematic issues remain in areas of isotropy13, homogeneity1415, curvature1617, symmetry1819, censorship20, inflation212223, expansion rate24, cosmological constant2526, dark matter2728293031, and dark energy3233 to name a few. In other early works, research noted infinite field solutions vs. finite field solutions and combinations thereof, Hoyle and Narliker solutions noted differences between Einstein’s Field Equations in the direct - particle interaction theory and those highly restrictive requirements34. The work presented here is in agreement with the Copenhagen interpretation of Heisenberg’s quantum mechanics35 using Dirac notation36 and by using methods introduced by Von Neumann which employ projection operators37 to show that a point source Singularity expanded to a folded 2-D Singularity surface via the mechanism of skeleton geometry and associated deficit angles possessing characteristics of folded geometry, infinite expansion, and infinite energy. Dirac notation and Newton-Raphson projection operators (attributed to Von Neumann) are used to model unique states of potential expansion of the 2-D Singularity surface in a way that supports the creation of a 3-D universe having the characteristics of vacuum, matter, energy, space. In the work presented, we build a framework of unique eigenstates to represent quantum energy transport phenomena. This framework is used to show that quanta (including ‘vacuum quanta’) can transport through an event horizon in a manner consistent with that which was first proposed by Hawking.38 We show that these eigenstates are represented by unique eigenvectors and eigenvalues that are probabilistic solutions to collapsed wave functions of Schrödinger’s wave equation for quanta observed in both nature and in interstellar space. Such evolutions of quanta out of the posterior side of the event horizon into the expanding vacuum in 3R could aide in the understanding of observed cosmic phenomena such as acceleration as seen in the very early Universe3940, inflation41, expansion rate42, whether the cosmological constant is actually constant434445, dark matter46, and dark energy47 and could help account for the numerous above mentioned observations and contemporary Cosmological model inconsistencies related to the distribution of matter and energy4849 which are based on Big Bang nucleosynthesis. We show that using a 2-D Singularity state of potential expansion and subsequent incremental changes to the state of potential expansion, eigenstate models are constructed of eigenvectors whose eigenvalues satisfy quantum energy transport phenomena out of the 2-D Singularity event horizon and to manifest as unique quanta. Such quanta include but are not limited to the range of Standard Model particles50 and Baryonic matter5152, photons of cosmic origins, and equally as important, vacuum quanta.

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Skeleton Geometry53 1st we must ask the questions: Can a Singularity be 2-Dimensional? And if so, what is the mathematical representation of such? Let us represent the point source Singularity (dimensionality of 1R) as an item of infinite energy and infinite density of radius r = 0 with an expansion potential = ∞. Let us define an event for Singularity as an expansion transition from 1R →2R by the smallest incremental change possible to substantiate a 2-geometry but which preserves the infinite energy at every point on 2R. At this 1st instantiation of geometry, we now have a manifold for an elementary flat-space building block of dimensionality 2 in the form of an arbitrary triangle consisting of a vertex and two scalar sides connected by a scalar base located opposite the vertex54 with the characteristic of infinite energy. Let us allow for a sufficient number of vertices to exist through expansion and allow for sufficiently small scalar lines connected at the vertices with sufficiently small scalar bases located opposite the vertices to form sufficiently small triangles having Euclidean geometrical properties to populate the entire 2-geometry and preserve the infinite energy. In Gravitation, we see how increasing the number of arbitrary paired lines about the vertex closely approximates a polyhedron built of triangles.55 We postulate that at the moment of Singularity expansion from 1R→2R, the state of potential expansion was acted on to produce a new state of potential quantum energy transport which preserved the infinite energy of the Singularity surface and which allowed for an event horizon posterior to the expansion through which all quanta including vacuum quanta could transport to be observed and measured. The expanded 2-geometry surface possesses characteristics of an anterior side that is unmanifested which contains a geometrized expansion potential in the plane along with the infinite energy of singularity at every point on the plane, and a posterior side possessing a state of potential for geometry and energy transport potential states which lead to a Relativistic event horizon through which quantized energy is manifested via mechanisms of uniquely characterized potential states of quantum energy transport. We also postulate that the anterior expansion rate cannot be known for reasons of nonobservability due in part because the event horizon prohibits any such observations, but also because nothing exists anterior to the expanding 2-D Singularity surface. [In our model, expansion can be observed and measured from any theoretical point posterior to the event horizon looking towards the cosmic expanding event horizon, thus the term 'posterior' meaning behind or remaining in the wake of the expansion vs. anterior meaning ahead of the surface of the ‘shell’ like 2-D Singularity expansion.] Although the 2-D expansion rate of the anterior surface cannot be known, an effect of the linear expansion of the posterior side of the 2-D singularity surface can be observed and measured. We see one such expansion effect noted by Silpher56 and later by Hubble who observed that the distances to faraway galaxies were strongly correlated with their spectral redshifts57, 58. We postulate that the expansion of the 2-D singularity surface results in characteristics of our

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expanding universe that include but are not limited to vacuum, space, geometry, energy, density, and time. Within the posterior volume now enclosed by the expanding 'shell', this expansion has been observed and the characteristics are consistent with an expanding universe.59, 60 In a review of Hawking's reference to Robertson's and Walker's metric61 for a model that is spatially homogeneous and isotropic {for edification, We include it herein as: ds2=dt2-R2 (t)[(dr2)/(1-Kr2)] +r2 (dθ2+sinθ dφ2) for K = 0 or +/-162, we can apply this metric to equations that describe a dynamic 2-geometry model for an expanding universe. Per Hawking63, the expanding 'steady-state' model consistent with Hubble64 is assumed to have started from a point source Singularity and which presents the same appearance at all times. The point source Singularity explosive expansion from 1R→3R is problematic. We argue that a potential state of expansion must 1st have to exist in 1R and for a radially outward explosive expansion to have occurred, would predispose that a medium for expansion would necessarily have to precede the Singularity explosion in order for the geometry and energy to follow; Which is contradictory to the notion that nothing existed prior to the Big Bang. We propose that a 2-D Singularity surface manifested out of a point source Singularity in 1R and expanded radially outwards from a state of potential expansion approximating a smooth continuous 2-D folded surface creating the universe on the posterior side of an event horizon. The 2-D Singularity surface from 1R→2R is continuing to propagate via a change mechanism from a state of potential expansion as a folded surface with non-Euclidian curvature akin to a radially expanding homogeneously and isotropically infinite energy dense spherical 'shell'. Folded Skeleton Geometry: An expanding 2-D surface can now be defined such that a deficit angle δ exists about an arbitrary vertex and between adjacent sides of Euclidean triangles. By connecting the new adjacent sides opposite the vertex created by the δ angle, a new triangle is created. Increasing the number of δs increases the number of triangles in the surface. If we assume these triangles to exist as a polyhedron surface expanding in a manner consistent with Hawking, the expansion of the 2geometry must necessarily create deficit angles between the arbitrary scalar sides of our triangles. Increasing the number of δs about the expanding surface and decreasing the scalar size of the angles, approximates a plane of smooth continuous 2-geometry. The sum of the deficit angles over all vertices has the value, 4π, as does the half integral of the continuously distributed scalar curvature taken over the entirety of the original smooth continuous surface: skeleton geometry ∑

δi = ½ ∫ actual smooth geometry (2)R d(surface) = 4π.65

Eq. 1 shows that a 2-geometry structure under conditions of curvature can fold on itself to become a smooth continuous surface of a sphere.

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(1)

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Generalizing from the n-geometry, Regge calculus approximates a smoothly curved, ndimensional Riemann manifold as a collection of n-dimensional blocks, each free of curvature and joined by (n-2) dimensional regions in which all the curvature is concentrated.66 Applying the Regge calculus to a 2-dimensional Riemann manifold in an expanding 2-D Singularity surface defined by triangles whose vertices are the δi, the δi are the “hinges” where all of the curvature in the 2-geometry is concentrated. Any two adjoining triangles can share a “hinge” and the “hinge’ is subject to folding. We will show that in Hilbert space67, an expanding 2-dimensional surface with a sufficient number of δs (deficit angles) between sufficient numbers of adjoining triangles will allow for a sufficient number of new “hinges” to allow folding to satisfy a smooth continuous boundary condition of area equal to 4π.

ANALYSIS OF AN EXPANDING 2-D SINGULARITY 2-D Singularity, Expansion, and Quanta Transport We begin by introducing Dirac notation to model concepts such as potential expansion and potential quantum energy transport, and operation of expansion and operations of quantum energy transport which are necessary in order to discuss our proposal. First, the operation of an event (in this case we take an expansion event) can be thought of as an operation which causes the state of potential to change. Expansion can be perceived as a change in the state of potential expansion |ψi⟩, by acting on the |ψi⟩. The result of this operation, i.e., |ψi+1⟩ = Ô expansion (here Ô represents the action of expansion, through an arbitrary operation which in general causes the change), is evaluated by comparing the changed state of potential expansion, i.e., |ψi+1⟩ with the previous state of potential expansion |ψi⟩. The same approach can be used to model potential states of quantum energy transport using alternative symbologies but the resulting form and functions are identical. If we assume our Singularity to have a potential function for expansion (and/or quantum energy transport)1, then it follows that we can define the potential for Singularity expansion herein as our |ψi⟩. We can also agree that the operation of Singularity expansion from the potential state |ψi⟩ to the next changed or expanded Singularity state can be represented as |ψi+1⟩. “1” is an integer in this case because the changed state of Singularity expansion cannot exist as a partial or fractional change. The potential state of Singularity expansion has either changed or it hasn’t and our index for identifying incremental potential state changes is arbitrarily chosen using integers. Let’s then postulate a sequence of pairs {|ψi⟩, Ci} consisting of a sequence of change events Ci during each of which some change operates upon and changes the state of potential |ψi⟩. The state |ψi⟩ constitutes a set of potentialities out of which the next change event Ci+1 arises. 1

Henceforth, we will limit our discussion in this section to states of potential expansion in order to avoid confusion, but the approach is equally valid for potential states of quantum energy transport and we will include those explanations later in our work.

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If there exists such a state of potential for expansion in a point source Singularity and there is a sequence of change events such as expansion, then we can represent such a potential for expansion and the sequence of expansion events as our set {|ψi⟩, Ci} and hence when an expansion event acts on the state of Singularity potential expansion, the result of the operation |ψi+1⟩ = Ô|ψi⟩ is evaluated by comparing the changed state of potential expansion |ψi+1⟩ with the previous state of potential expansion |ψi⟩. This process of change creates an actual event in Singularity expansion and it becomes a phenomenon of expansion through the operation of expansion. Let us now represent the state of potential expansion by a vector in Hilbert space. Using the notation, we can write a state vector as |ψ⟩ which is a linear combination of the basis vectors |i⟩ with i = 1, 2, 3,…, N, namely as follows:

|ψ⟩ = ∑Ni=1 ψi |i⟩

(2)

The basis vectors |i⟩ give all possible states describing events for any potential state in question. The N vectors together form a complete basis set of states, namely all potential outcomes in any domain (finite or infinite). The discrete variable i as a label for the basis elements can be a continuous variable; in which case Eq 2 can be replaced by integration. Because expansion in a Singularity needs to carry out an operation of expansion inside this space, to make the expansion event happen, this vector space should have the property of some finite measure, even if it is infinitesimally small, and as a result it is a Hilbert space. In this Hilbert space, the measure of the overlap between two states |ψ⟩ and |ф⟩ is measured by the scalar product between the two vectors representing the two states namely:

⟨ψ|ф⟩ = ∑i ψi*фi

(3)

It is important to note that the overlap of any state to itself being the square of the length of the vector, is normalized to unity which is ⟨ψ|ψ⟩ = 1, which is possible when in Hilbert space. Furthermore, if we allow our basis vector |i⟩ to consist of Singularity expansion elements on our 2-D Singularity surface to be the deficit angles δi, our state vector for potential expansion |ψ⟩ using Eq 2 becomes:

|ψ⟩ = ∑Nδ=1ψδ |δ⟩

(4)

Which then allows us to write |ψi+1⟩ = Ô|ψi⟩ as |ψδ+1⟩ = Ô|ψδ⟩. In Eq 1, we showed that the 2-D singularity surface consisted of deficit angles δ with sufficient numbers and decreased sizes to have an area = 4π. Using deficit angles of sufficient size and number on our expanding 2-D singularity surface, the index N in Eq 4 becomes ∞

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(infinite). Replacing N with ∞ into Eq 4 we can write our state vector for potential expansion as:

|ψ⟩ = ∑ꝏδ=1 ψδ |δ⟩

(5)

The vectors |δ⟩ give all possible states of expansion (states describing potential expansion events). Because expansion needs measurable operations of expansion inside this space, to make an expansion event happen, this vector space should have the property of some finite measure and as a result of these requirements, our 2-D Singularity surface expansion can be said to take place within a Hilbert space. We can safely say that |ψ⟩ represents the state of potential expansion which is unmanifested. The event of expansion can be mathematically broken down into a two-step process: a) the operation which applies a change on the state of potential expansion and that transforms the state of potential expansion to a state representing the expansion itself (|ф⟩ = Ô|ψ⟩ where Ô is the operator representing a particular action of expansion) ; and b) the overlap of the expanded state (after the operation of expansion) to the state of potential expansion prior to the expansion, i.e. M = ⟨ψ|ф⟩, corresponds to the expansion value. After this process in Hilbert space, the expansion becomes real and measurable. As an example, the Newton-Raphson operator Ô creates the potential solution of an equation. When the state |xn+1⟩ = Ô|xn⟩ and the previous state |xn⟩ have large overlap we take it that the solution is measurable (or “observed”). This is how we decide that we have found the solution, namely when ⟨xn|Ô|xn⟩ = 1. To verify we have found an expansion independent of the initial state, we may start from the state |ψ0⟩ = ∑Mi=1 |ψ 10 ⟩ and after application of the operator Ô several times, we stop when the overlap ⟨ψn|Ô|ψn⟩ is maximum (or unity if we keep normalizing states |ψn⟩). Consequently:   

Every action of expansion can be mathematically represented by an operator Ô applying expansion to the state of potential expansion. This causes a change in the state of potential expansion. This changed state of potential expansion |ф⟩ = |ψ0⟩, due to the operation of expansion remains in a state of potentiality until it expands through the operation of expansion. The new action of expansion which causes the expansion to be measured by comparison of the two states, namely the one before the operation of expansion, i.e. |ψ⟩, with the one after the operation of expansion, i.e., Ô|ψ⟩, which is taken to be the scalar product between the two states:

M = ⟨ψ|Ô|ψ⟩

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(6)

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

The result of this comparison is consistent with above discussions. Also, when this operator is used in Hilbert space to represent a real (as opposed to imaginary/complex) expansion, i.e., M=M*, the operator Ô is a Hermitian operator.68 Each particular operation of expansion represented by an operator Ô that represents a particular expansion operation, is characterized by eigenvectors and eigenvalues in Hilbert space, namely

Ô|λ⟩ = λ|λ⟩ 

(7)

The significance of the eigenvectors of Ô is that these are the only states of potential expansion that do not change by the particular act of an expansion event. The result of the event is the corresponding eigenvalue because the projection of the result of the expansion, i.e., Ô|λ⟩ on the state |λ⟩ itself, is the eigenvalue λ. The eigenstates are the only states which represent a measureable event in expansion through the event which corresponds to the eigenvalue.

Let us consider two such operators, the operator Ô and its eigenstates/eigenvalues as defined above, and the operator Q̂ with the following spectrum of eigenstates/eigenvalues:

Q̂ = |μ⟩ = μ|μ)

(8)

Since the eigenstates of each of these operators form a complete basis set of a Hilbert space, let us state the eigenstates of the operator Ô in terms of the operator Q̂, namely:

λ⟩ = ∑μ ψλ(μ)|μ)

(9)

ψλ(μ) = μ|λ).

(10)

To apply this expression to an event, let us suppose that the event represented by the operator Ô transforms the state of potential expansion to a particular eigenstate |λ⟩. The result of the event is the corresponding eigenvalue λ. We do not assume that the next expansion event is in any way related to the previous expansion event and we represent the next expansion event by the operator Q̂. The result of a single expansion observation corresponding to Q̂ will transform the state of potential expansion to an eigenstate |μ) of Q̂ corresponding to a measured expansion characterized by the eigenvalue μ. The result of a single event will bring about in expansion only a single definite answer. We argue that this answer must necessarily correspond to an eigenstate of the operator Q̂ because only the eigenstates of an operator are ‘real’ or ‘valid’ against the application of Q̂, and we can successfully state that any arbitrary state of potential expansion must necessarily have dedicated eigenstates represented by unique eigenvectors whose solution elements are also unique expansion eigenvalues.

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As eigenstates correspond to arbitrary expansion events, let us represent a body of arbitrary expansion operators Ɵ̂ on our 2-D singularity surface by:

∑

ꝏ

i=1

Ɵ̂i

(11)

With the following spectrum of eigenstates/eigenvalues:

Ɵ̂i = |μi⟩ = μi| μi)

(12)

Such that the eigenstates of each of these operators in Ɵ̂ form basis sets in Hilbert space, let us represent the eigenstates ƴ⟩ of the operators for Ɵ̂i using the above approach in Eq’s 9 and 10 in terms of: ƴi⟩

= ∑ƴi ψƴi (μi)| μi)

(13)

ψƴi (μi) = μi|ƴi).

(14)

Using the results from Eq 4 and substituting the Singularity expansion basis vectors |ƴi⟩ with our deficit angles δi, Eq’s 13 and 14 become

ƴδi⟩ = ∑δiψδi (μδi)| μδi)

(15)

ψδi (μδi) = μδi|ƴδi).

(16)

As the deficit angles δi approach zero while the number of deficit angles δi approaches infinity, the 2-D Singularity surface approximates that of a smooth continuous surface. As a consequence of our approximation of a smooth continuous surface, the ‘hinges’ where all of the geometry is concentrated must be homogeneously and isotropically distributed throughout the entire 2-D Singularity surface. With that in mind, we can successfully conclude that the 2-D Singularity surface approximates a spherical surface. And integrating across the entire 2-D Singularity surface deficit angles δi, Eq 15 and Eq 16 can be re-written as:

∫0i=1 ƴδi⟩dδ

=

∫0i=1 ψδi (μδi)| μδi) dδ

Discussion of the Main Results

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=

4π.

(17)

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We have constructed a model of an expanding point source Singularity through an expansion modulus, namely a state of potential expansion that undergoes expansion. We showed that when the state of potential expansion changes to an expanded state, deficit angles can be used to represent the changes in expansion. The vertices of the deficit angles were shown to be the concentrations of geometry which allow for expansion of a 2-D surface to fold upon itself while maintaining a surface area of 4π. We argue that if such an expanding 2-D Singularity surface exists, one property it would possess is that Singularity must necessarily exist at each and every point on the surface. Thus, any point on the 2-D Singularity surface can be represented by an arbitrary state of potential expansion, and that collectively, we can represent any and all points on our 2-D Singularity surface as arbitrary states of potential expansion subject to arbitrary operations of expansion. Let us define a point source Singularity of infinite energy density in 1R with a state of potential expansion |ф⟩ that undergoes a change in the state of expansion to a new state of expansion into 2 R that has new states of potential expansion. Using the results from Eq’s 7 through Eq’s 16 let us state the transition for our 1R Singularity into 2R as:

{1R: [|ф⟩, C] →2R: Ô|ψi⟩ → ⟨ψn|Ô|ψn+1⟩ |ψ⟩ → ∑ꝏδ=1 ψδ |δ⟩ = 4π}.

(18)

We further define that any change in any state of potential expansion on the 2-D Singularity surface leads to curvature consistent with deficit angles defined above. Let us assume our 2-D Singularity surface approximates a smooth continuous surface, and using the results of Eq 17, let us define our 2-D surface of Singularity in 2R such that the surface consists of energy density that is infinite at any point anywhere and that any point on the plane contains a unique potential state of expansion. As such, the entire surface of our 2-D Singularity is populated with an infinite number of unique states of potential expansion. Let us also assume for simplicities sake that the changes to the unique states of potential expansion happen simultaneously and result in a folded surface which approximates a spherical shape. We feel safe in this assumption because to argue that simultaneity does not exist is to imply that a ‘time’ function exists within the confines of our 2-D Singularity folded surface; i.e. a state of potential time must exist. In our model, we allowed for a state of potential expansion only. We excluded a state of potential time. If such a potential time function were to exist within a Singularity, that would necessarily obviate the conditions for infinite energy density, since by definition a Singularity is void of time. Such a function would also give rise to an infinite number of expansion shapes to our folded surface (including discontinuities) as opposed to one that approximates a smooth continuous spherical shape. Let us further define the 2-D Singularity spherical surface such that the surface contains two planes: an outer side that is unobservable and unmanifested in which all of the expansion and folded geometry is concentrated. For the sake of reference we shall call this the convex or ‘anterior’ side of the folded 2-D spherical surface. If a vector could exist on this anterior (unmanifested) side of our 2-D Singularity spherical surface, the vector would point in a direction normal to a tangent plane on the spherical surface.

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Given the definition of ‘anterior’, let us define an inner side for our 2-D Singularity surface. For the sake of reference, we shall call this the concave or ‘posterior’ side of the folded 2-D spherical surface. If a vector could exist on this posterior side of our 2-D Singularity spherical surface (admittedly, this Singularity surface could be referred to without objection as a ‘membrane’), the vector would point in a direction normal to a tangent plane on the convex surface. This vector would also point towards the theoretical location of the point source Singularity. We argue this is the side that begins to manifest Lorentzian characteristics of infinite energy density such as temperature, vacuum, curvature, energy transport phenomena, etc. On our expanding 2-D Singularity surface, we showed that arbitrary states of potential expansion can have unique representations in Hilbert space and that these unique states of potential expansion can have unique eigenvectors that consist of unique expansion basis vectors whose solutions are eigenvalues that are the unique deficit angles. Let us now emphasize that the result of an event represented by an operator ‘Ô’ transforms the state of the potential event to a particular eigenstate λ. The next event is represented by Q̂ which may or may not be compatible with the previous event Ô. The result of a single observation corresponding to Q̂ will transform the state of potential event to an eigenstate |μ) of Q̂ corresponding to a definite event characterized by the eigenvalue μ. The result of a single event will bring about only a single definite answer. This answer must correspond to an eigenstate of the operator Q̂ because only the eigenstates of an operator are valid against the application of Q̂. As we have already mentioned, this is the reason why we use eigenstates to represent any particular realizable state of either expansion, or a particular realizable state of quantum energy transport phenomena.

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Quantum Energy Transport out of a 2-D Singularity Plane Using the above Hilbert space approach as a framework, let us define a quantum energy transport mechanism from the 2-D Singularity surface to a point posterior of the event horizon as a change in a state of potential quantum energy transport using the state vector |χ⟩ which is a linear combination of the basis vectors |i⟩ with i =1,2,…,N, namely, as follows:

|χ⟩ = ∑Ni=1 χi |i⟩.

(19)

The vectors |i⟩ give all possible states of quantum energy transport (states describing potential quantum energy transport) for some arbitrary quanta in question. All the N vectors together form a complete basis set of states, namely, they cover all potential quantum energy transport outcomes. It is important to note that N can be either finite or infinite, and if infinite, would imply an infinite number of potential quantum energy transport mechanisms may exist. As such, the discrete variable i, labeling the basis elements, can be a continuous variable; in this case the summation in Eq 19 is intregable over ꝏ. The above linear combination implies that the result of any quantum energy transport is not in any of the potential states of quantum energy transport. Unless observable events of quantum energy transport take place, the best we can argue is that there are discrete states of quantum energy transport potentialities. We argue that this is so - not because we do not know what the actual measureable value of the quantum energy transport is, - we argue it is so because there is no value in the act of quantum energy transport. To support this, one could argue what is the potential observation of the quanta before observing it? The state vector which is represented as a linear combination of potential quantum energy transport events represents the state of potential quantum energy transport not events in quantum energy transport. It is through the operation of quantum energy transport that one of the potential quantum energy transport events can take place. Because of the potential nature of the state of quantum energy transport (i.e., that which the state of potential quantum energy transport describes, is not actual yet before the event) it is written as a mixture of possibilities. Each possibility is unique and distinct from any other. The result of the event while unique, prior to the event itself (when it is in potentia), should be written in such a way that it is a mixture or a sum of probability amplitudes for each quantum energy transport event to occur as opposed to just probabilities. The reason is that we need to end up with probabilities after the event not prior to the event. This is so in order to allow for the operation of quantum energy transport to take place and then carry out the event of the quantum energy transport by comparing the previous state of potential quantum energy transport with the state of quantum energy transport after the operation in order to have a measurable event. After this quantum energy transport phenomena, we end up with real measurable quantum energy events on the posterior side of the event horizon with a probability given by the square of

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the coefficient in the linear combination multiplying the particular quantum energy state that becomes manifest. We argue that ‘real measurable quantum energy events’ are on the posterior side of the event horizon simply because no quantum energy events can be made anywhere between the 2-D Singularity surface and the event horizon. This also implies that all measureable quantum energy events (quanta events) take place within the geometry of the folded 2-D Singularity surface event horizon ‘envelope’, i.e., the volume region on the posterior side of the 2-D Singularity surface bounded on all sides by the event horizon. Using Eq 19 and our use of Hilbert space, let us define a vector for the all possible states of quantum energy transport as |ψ⟩ (states describing potential quantum energy transport events). Because quantum energy transport events need measurable operations of quantum energy transport inside Hilbert space, to make a quantum energy transport event happen, this vector space should have the property of some finite measure. And as a result of these requirements, our quantum energy transport can be said to take place within a Hilbert space. As above, we can again say that |ψ⟩ represents the state of potential quantum energy transport. The event of quantum energy transport can be mathematically broken down into the two-step process from above: a) the operation which applies a change on the state of potential quantum energy transport and that transforms the state of potential quantum energy transport to a state representing the quantum energy transport event (|ξ⟩ = Θ̂|ψ⟩ where Θ̂ is the operator representing a particular action of quantum energy transport) ; and b) the overlap of the quantum energy transport state (after the operation of quantum energy transport) to the state of potential quantum energy transport prior to the quantum energy transport, i.e. M = ⟨ψ|ξ⟩, corresponds to the quantum energy transport value. After this process, the quantum energy transport becomes real and measurable. To verify we have found a quantum energy transport event independent of the initial state, we may start from the state |ψ0⟩ = ∑Mi=1 |ψ10 ⟩ and after application of the operator Θ̂ several times, we stop when the overlap ⟨ψn| Θ̂|ψn⟩ is maximum (or unity if we keep normalizing states |ψn⟩). To verify we have found a quantum energy transport event independent of the initial state, we may start from the state |ψ0⟩ = ∑Mi=1 |ψ10 ⟩ and after application of the operator Θ̂ several times, we stop when the overlap ⟨ψn| Θ̂|ψn⟩ is maximum (or unity if we keep normalizing states |ψn⟩). And as before: 



Every action of quantum energy transport can be mathematically represented by an operator Θ̂ applying quantum energy transport to the state of potential quantum energy transport. This causes a change in the state of potential quantum energy transport. This changed state of potential quantum energy transport |ξ⟩ = |ψ0⟩, due to the operation of quantum energy transport remains in a state of potentiality until it transports through the event horizon via an operation of quantum energy transport.

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

The new action of quantum energy transport which causes the quantum energy transport to be measured by comparison of the two states, namely the one before the operation of quantum energy transport, i.e. |ψ⟩, with the one after the operation of expansion, i.e., Θ̂|ψ⟩, which is taken to be the scalar product between the two states:

M = ⟨ψ|Θ̂|ψ⟩



The result of this comparison is consistent with above discussions. Also, when this operator is used in Hilbert space to represent a real (as opposed to imaginary/complex) quantum energy transport, i.e., M=M*, the operator Θ̂ as shown above is a Hermitian operator.69 Each particular operation of quantum energy transport represented by an operator Θ̂ that represents a particular quantum energy transport operation, is characterized by eigenvectors and eigenvalues in Hilbert space, namely

Θ̂|ϒ⟩ = ϒ|ϒ⟩ 

(20)

(21)

The significance of the eigenvectors of Θ̂ is that these are the only states of potential quantum energy transport that do not change by the particular act of quantum energy transport. The result of the event is the corresponding eigenvalue because the projection of the result of the quantum energy transport, i.e., Θ̂|ϒ⟩ on the state | ϒ ⟩ itself, is the eigenvalue λ. The eigenstates are the only states which represent a measureable event in quantum energy transport through the event which corresponds to the eigenvalue.

Let us consider two such operators, the operator Θ̂ and its eigenstates/eigenvalues as defined above, and the operator Ω̂ with the following spectrum of eigenstates/eigenvalues:

Ω̂ = |ν⟩ = ν|ν)

(22)

Since the eigenstates of each of these operators form a complete basis set of a Hilbert space, let us state the eigenstates of the operator Θ̂ in terms of the operator Ω̂, namely: ϒ⟩

= ∑ν ψϒ (ν)|ν)

ψϒ(ν) = ν|ϒ)

(23) (24)

To apply this expression to an event of quantum energy transport, let us suppose that the event of the quantum energy transport represented by the operator Θ̂ transforms the state of potential quantum energy transport to a particular eigenstate |ϒ⟩. The result of the event is the corresponding eigenvalue ϒ.

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We do not assume that the next quantum energy transport event is in any way related to the previous quantum energy transport event and we represent the next quantum energy transport event by the operator Ω̂. The result of a single quantum energy transport observation corresponding to Ω̂ will transform the state of potential quantum energy transport to an eigenstate |ν) of Ω̂ corresponding to a measured quantum energy transport characterized by the eigenvalue ν. The result of a single event will bring about in quantum energy transport only a single definite answer. We argue that this answer must necessarily correspond to an eigenstate of the operator Ω̂ because only the eigenstates of an operator are ‘real’ or ‘valid’ against the application of Ω̂, and we can successfully state that any arbitrary state of potential quantum energy transport must necessarily have dedicated eigenstates represented by unique eigenvectors whose solution elements are also unique quantum energy transport eigenvalues. As eigenstates correspond to arbitrary quantum energy transport events, let us represent a body of arbitrary quantum energy transport operators Θ̂ on our 2-D Singularity surface by:

∑

ꝏ

i=1

Θ̂i

(25)

With the following spectrum of eigenstates/eigenvalues:

Θ̂i = |νi⟩ = νi|νi)

(26)

Such that the eigenstates of each of these operators in Ɵ̂ form basis sets in Hilbert space, let us represent the eigenstates ϒ⟩ of the operators for Θ̂i using the above approach in Eq’s 23 and 24 in terms of: ϒ i⟩

= ∑ν i ψϒi (νi)| νi)

ψϒi (νi) = νi|ϒi).

(27) (28)

Therefore, we can state that Eq 27 and Eq 28 represent the unique eigenstates that must necessarily map to unique eigenvectors and unique eigenvalues of unique quanta (i.e. photons and/or particles) found within the envelope bounded by the event horizon located on the posterior side our expanding folded 2-D Singularity surface. Let us explore the meaning of Eq 27 and Eq 28 as follows: 1st let us suppose that the event of the quantum energy transport represented by the operator Θ̂ transforms the state of potential quantum energy transport to a particular eigenstate |ϒi⟩. The result of the event is the corresponding eigenvalue ϒi. The next observation is represented by Ω̂, which may or may not be compatible with the previous observation Θ̂. The result of a single observation corresponding to Ω̂ will transform the state of potential quantum energy transport to an eigenstate |νi) of Ω̂ corresponding to a quantum energy transport event characterized by the eigenvalue νi. The result

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of a single quantum energy transport event will bring about in quantum energy transport only a unique quantum energy transport event. This event must correspond to an eigenstate out of the operator Ω̂ because only the eigenstates of the operator Ω̂ are unique against the application of Ω̂. To repeat, this is the reason why we use eigenstates to represent any particular realizable state of potential quantum energy transport. The particular state |νi) which would be brought about in quantum energy transport cannot be known, all that is known is that the previous state of potential quantum energy transport is |ϒi⟩. We can also argue that while the state of potential quantum energy transport is |ϒ⟩, i.e., an eigenstate of the observable represented by the operator Θ̂, quantum energy transport carries out an event of an observable represented by the operation Ω̂. A particular question can be the following Pν : “Is the state of potential quantum energy transport the one corresponding to the eigenvalue ν?” This question is operationally applied using the projection operator defined as follows:

P̂ν|ν′) = δν ν′|ν′); [please note that δ is not to be confused with the δ which earlier denoted our deficit angles]

(29)

i.e., such that the outcome of its operation on the state |ν′) and then projected back to itself (measured against itself) is given as

(ν′|P̂ν|ν′) = δν ν′.

(30)

Namely, it is affirmative or negative depending on whether or not the state of potential quantum energy transport agrees with that sought by means of the operational question Pν. If the same question is applied on the state |ϒ⟩ given by Eq. 28, we find

⟨ϒi |P̂νϒi ⟩ = |ψϒi (νi)|2,

(31)

Namely, the outcome of the projection would be the eigenstate |ν) with eigenvalue ν and with probability |ψλ(ν)|2. Therefore, we can represent the projection operator in Hilbert space as P̂ν = |ν)(ν|.

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Characteristics of Expansion, Time, and Quantum Energy Transport out of a 2-D Singularity Plane Given that we theorize characteristics of phenomena occurring on a 2-D Singularity surface to include expansion events based on the mathematical model of skeleton geometry where all the geometry in concentrated in the ‘hinges’ that allow for geometrical folding through deficit angles (our δis), let us characterize elements of observed phenomena on the posterior side of the event horizon to include a chronology or sequence of change events. In the physical universe, fluctuation is inherent to observed phenomena of change. In quantum energy physics phenomena, we create measurement metrics in order for us to perceive and measure incremental changes of one energy state change to a subsequent energy state. One basis for our measurement metrics is a frequency based clock whose periodic changes or fluctuations are fundamental elements of quantum energy transport. We perceive time only through the direct perception of changes through an event. The value of the time interval between two successive events in quantum energy transport is only found by counting how many fluctuations of a given periodic event took place during these two events. Therefore the notion of time is related to the sequential (ordered) events which allow counting, and the interval of time and change (in particular periodic change) are complementary elements and they are not independent of each other.70 There is physiological evidence suggesting the direct perception of frequency. For example, we perceive the frequency of sound directly as notes or pitch, without having to perceive time and understand intellectually (after processing) that it is periodic. Other evidence of direct perception of frequency comes from color being perceived directly without the requirement that or any co-experience of time whatsoever. In addition, the retina receptor cells are highly sensitive and it has been shown that they can observe a single photon71 72. Furthermore, in biological systems, receptors for what we refer to as time do not exist.73 On the contrary, there is significant neuro-physiological evidence that the perception of time takes place via coherent neuronal oscillations74 which bind successive events into perceptual units75. The physical universe responds to frequency very directly, and some examples are resonance, single photon absorption, and in general absorption at definite frequency. The timeless photon, in addition to being a particle, can be thought of as the carrier of the operation of quantum energy transport on the state of potential quantum energy transport. When an operated state of unique potential quantum energy transport is measured against its own unique state before the operation, a definite frequency is realized (or manifested). An instrument (such as the photon counter) is needed to quantify (manifest) the operation of quantum energy transport, because matter is the necessary “mirror” to “reflect” (to manifest) the act of quantum energy transport. At first glance it may appear that we have introduced a duality by separating quantum energy transport and matter. Matter, we argue, is uniquely manifested energy in another unique state of potential quantum energy. Let us try to discuss evolution (or change) quantitatively. In order to describe any perception of change, we invent a parameter which we call time which labels the various phases of change and

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we share a delusion belief amongst others that such a parameter has independent existence from quantum energy transport. We argue that time is only a device invented to measure change and to facilitate the description of changes. Therefore, we imagine the state of potential quantum energy transport |ψ(t)⟩, as a function of t, labeling the time of potential observation carrying forward from our earlier work all of the associated uniquenesses of each and any function involving |ψ⟩, |ψ(t)⟩, etc. We wish to discuss a periodic motion, so let us confine ourselves within a cycle of period T. For simplicity we will discretize time, namely, the states are labeled as |ψ(ti)⟩ where t1 = 0, t2 = δt, t3 = 2δt,...,tN = (N − 1)δt, with Nδt = T. (Again note that our δ is not to be confused with our deficit angle δ.) These time labels have been defined and measured in terms of another much faster periodic change which we call ‘a clock’. Let us assume that δt corresponds to the “time” T′ of a single period of the fast periodic change of the clock, namely ti, i = 1, 2, … , N, are the moments when the “ticks” of the clock occur. Let us define the chronological operator t̂ and its eigenstates t̂|ψ(t)⟩ = t|ψ(t)⟩,

(32)

namely, we have assumed that the state of potential quantum energy transport is characterized by a definite measured time. Notice, that we needed two periodic motions “running” in parallel in order to discuss the measurement of the period of the first in terms of the second (clock). Namely, we are unable within a single event in quantum energy transport to know both the time and the frequency of the event. We have already discussed that physiological evidence given above, suggests that quantum energy transport only experiences frequency not time as a fundamental characteristic. Note that time here is only quantified through the periodic motion. Let us now define the eigenstates characterized by definite periods in terms of the states characterized by definite time. We define the evolution operator or time-displacement operator, the operator that causes the change of the state of potential quantum energy transport, namely T̂|ψi⟩ = |ψi+1⟩, i=1, ... , N−1, T̂|ψN⟩ = |ψ1⟩,

(33)

where |ψi⟩ = |ψ(ti)⟩, and the second equation above implies that because of the nature of the perception of the periodic change there will be no difference in the state after time t = Nδt, i.e., the period T of the periodic change. In the case of periodic change, such as described by Eq. 33, all the eigenstates and eigenvalues of the operator T̂, in terms of the eigenstates of the chronological operator, are given as follows:

T̂|ωn⟩ = τωn|ωn⟩, τωn = e−iωnδt, Pg 18 of 25

(34)

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|ωn⟩ =

1

/√N ∑

N i=1

e n i |ψi⟩, iω t

(35)

where ωn = n(2π/T ), and n = 1, 2, ..., N . Notice that the quantization of the levels of single periodic change is the same as that of the harmonic oscillator (here we have used natural units where the so-called “energy” is the same as the frequency). This is so because we have not limited in any way the number of quanta which are observed. The state which describes a periodic change is such that when the time displacement operator acts on it, it behaves as its eigenstate. Namely, they are the only states of potential quantum energy transport which do not change under observation (or measurement), no matter how many times the observation (measurement) occurs. Therefore, the measurement in this case introduces no frequency uncertainty of the state of definite frequency. This state cannot be characterized by any definite value of time; for the change to be characterized by a definite frequency, an observation of regular periodic motion is required to continue forever. Time t = mδt “elapses” when the time-translation operator T̂ acts m consecutive times on the state, namely, the time evolution of the state |ωn⟩ is |ωn⟩t = T̂m|ωn⟩ = =

e-

|ωn⟩

iω mδt

n

e- n |ωn⟩ iω t

(36) (37)

Let us now consider the case where the change does not necessarily occur at a single period but it is a mixture of periodic changes of various characteristic frequencies. We begin from the frequency eigenstates Eq. 35 as the basis and let us define the time evolution of the state as |ψ(t)⟩ =

∑n cn|ω ⟩

(38)

∑n cn e- n |ωn⟩,

(39)

=

n t

iω t

where the sum is over all the eigenstates of the time translation operation acting on the state of potential quantum energy transport that characterizes the system. This latter equation can also be written as follows

e-

|ψ(t)⟩ = iω̂nt |ψ(0)⟩, ω̂|ωn⟩ = ωn|ωn⟩,

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(40) (41)

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where |ψ(0)⟩ is some reference state. Again the perception of frequency is direct and so in our description of nature we need to start by considering this perception as one fundamental building block of consciousness and not the perception of time. Equivalently from Eq. 19, we can say that this is the solution to the following differential equation

ω̂ |ψ(t)⟩ = i∂t|ψ(t)⟩,

(42)

ω̂ = i∂t

(43)

or equivalently

We need to discuss why the above frequency operator characterizes the measurement of change. For simplicity let us go back to the discrete time domain. If quantum energy transport applies the operator ω̂ on the state of potential quantum energy transport we have

ω̂|ψ(ti)⟩ =

i

/ δt (|ψ(t

i+1)⟩−|ψ(ti)⟩),

(44)

which is (apart from the multiplicative factor i/δt) the change of the state of the unique potential quantum energy transport. This change is evaluated by simply using a measure of the instantaneous state of unique potential quantum energy transport itself, i.e., by projecting the change onto |ψ(ti)⟩. This means that the expectation value ⟨ψ(t)|ω̂|ψ(t)⟩ is a measurement of the rate of change of a unique potential quantum energy transport. Using Eq. 43 for the frequency operator, the following commutation relation between frequency and the chronological operator follows in a straightforward manner:

[t̂, ω̂] = i.

(45)

In addition, the well-known uncertainly relation follows, namely,

ΔωΔt ≥ 1.

(46)

The uncertainty relationship between frequency and time can be easily understood as follows. Let us suppose that a changing state of quantum energy transport is observed for a finite interval of time Δt. This observation time interval is also the uncertainty in time, because there is no particular instant of time inside this interval to choose as the instant at which the observation of the event happened. Even if the event seems to be regular or periodic inside this interval of time, there is an uncertainty as to what happens outside this interval. In fact, as discussed, nothing happens outside this interval because there is no observation in quantum energy transport and thus no event there, only a unique potentiality. If we calculate the Fourier spectrum of such a changing event, no matter how regularly it evolves inside this interval, (with nothing happening

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outside of this interval) we will find significant amplitudes for frequencies in an interval range greater than 1/Δt.

TBS – WIP In the next section, we argue that our expanding 2-D Singularity surface is a modality for an expanding universe consistent with conclusions shown by Einstein76, Hawking77, Wheeler78, and others but absent of the many inconsistencies noted above.

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TBS - WIP Gravitational Theory We agree, as stated in Gravitation, that the convex (posterior) surface advancing away from a theoretical center can be considered ‘locally Euclidean’. We also respect the significance of spacetime geometry as locally Lorentzian everywhere. 79

The difficulty was solved when it was realized that the universe was expanding, since in an expanding universe the retarded solution of the above equation is finite by a sort of ‘redshift' effect. The advanced solution will be infinite by a 'blue-shift ' effect. This is unimportant in Newtonian theory, since one is free to choose the solution of the equation and so may ignore the infinite advanced solution and take simply the finite retarded solution. Solutions to the Einstein Field Equations show us how energy density creates curvature.80

The question remains, however, as to how the infinite energy density at Singularity would not instantly create a self-annihilating curvature going from 1R →2R.

TBS – WIP 1

Foundations of Physics Publication date: June 6, 2006 (Found. Phys. 36 (6)) Published on line: DOI: 10.1007/ s10701-006-9049-9 http://dx/doi.org/10.1007/s10701-006-9049-9 2 Einstein, Albert (1916). "The Foundation of the General Theory of Relativity"; Annalen der Physik. 354 (7): 769. Bibcode: 1916 AnP...354.769E. doi:10.1002/andp.19163540702. Archived from the original (PDF) on 2012-02-06. 3 Hawking, Stephen; Penrose, Roger; (1970). "The Singularities of Gravitational Collapse and Cosmology". Proceedings of the Royal Society A. 314 (1519): 529–548. Bibcode: 1970RSPSA.314.529H. doi:10.1098/rspa.1970.0021 4 ibid; Einstein 5 Penrose, Roger (1965), "Gravitational collapse and spacetime singularities", Physical Review Letters, 14 (3): 57– 59, Bibcode: 1965 PhRvL..14...57P, doi:10.1103/PhysRevLett.14.57 6 Hawking, S. W.; Penrose, R.; (1970); "The Singularities of Gravitational Collapse and Cosmology", Proc. R. Soc. A, 314 (1519): 529–548, Bibcode: 1970RSP SA.314..529H, doi:10.1098/rspa.1970.0021 7 Penrose, R. 1965, “Gravitational collapse and space-time singularities”, Phys. Rev. Lett., 14, 579 8 Penrose, R. 1969, “Gravitational collapse: the role of general relativity, Rivista del Nuovo Cimento”, Numero speciale, 1, 252–276. 9 Hawking, S. W., Penrose, R. 1970; The Singularities of Gravitational Collapse and Cosmology, Proc. R. Soc. (London), A314, 529-548. 10 Luminet, Jean-Pierre (2008). The Wraparound Universe. CRC Press. p. 170. ISBN 978-1-4398-6496-8. Extract of p 170. 11 Hetherington, Norriss S. (2014). Encyclopedia of Cosmology (Routledge Revivals): Historical, Philosophical, and Scientific Foundations of Modern Cosmology. 12 Novello, M., Perez Bergliaffa, S. E.: Bouncing cosmologies. Phys. Rept. 463, 127 (2008) 13 M. Lachieze-Rey; J.-P. Luminet (1995), "Cosmic Topology", Physics Reports, 254 (3): 135–214, arXiv:grqc/9605010 Freely accessible, Bibcode:1995PhR...254..135L, doi:10.1016/0370-1573(94)00085-H

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Hawking, Stephen W.; Ellis, George F. R. (1973); The Large Scale Structure of Space-Time, Cambridge University Press, ISBN 0-521-09906-4 pp. 351ff. The original work is Ehlers, J., Geren, P., Sachs, R.K.: Isotropic solutions of Einstein-Liouville equations. J. Math. Phys. 9, 1344 (1968). For the generalization, see Stoeger, W. R.; Maartens, R; Ellis, George (2007), "Proving Almost-Homogeneity of the Universe: An Almost Ehlers-Geren-Sachs Theorem", Astrophys. J., 39: 1–5, Bibcode:1995ApJ...443....1S, doi:10.1086/175496 15 Ellis, G. F. R.; MacCallum, M. (1969). "A Class of Homogeneous Cosmological Models". Comm. Math. Phys. 12 (2): 108–141. Bibcode:1969CMaPh..12..108E. doi:10.1007/BF01645908 16 McCrea, W. H.; Milne, E. A. (1934). "Newtonian Universes and the Curvature of Space". Quarterly Journal of Mathematics. 5: 73–80. 17 Guth, Alan H. "Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems". Physical Review D. 23 (2): 347–356. Bibcode:1981PhRvD..23..347G. doi:10.1103/PhysRevD.23.347 18 P. Ojeda and H. Rosu (2006), "Supersymmetry of FRW barotropic cosmologies", International Journal of Theoretical Physics, 45 (6): 1191–1196, arXiv:gr-qc/0510004 , Bibcode:2006IJTP...45.1152R, doi:10.1007/s10773006-9123-2 19 Weinberg, S. (1976). "Implications of Dynamical Symmetry Breaking". Phys. Rev. D. 13 (4): 974–996. 20 Τ. P. Singh; J. Astrophys. Astr. (1999) 20, 221–232 Gravitational Collapse, Black Holes and Naked Singularities 21 Guth, Alan H. (1997), The Inflationary Universe, Reading, Massachusetts: Perseus Books, ISBN 0-201-14942-7 22 The Self-Reproducing Inflationary Universe by Andrei Linde, Scientific American, Volume 9, Issue 1 (1998) 98104. 23 Starobinsky, A. A. (December 1979). "Spectrum Of Relict Gravitational Radiation And The Early State Of The Universe". Journal of Experimental and Theoretical Physics Letters. 30: 682. Bibcode:1979JETPL..30..682S.; Starobinskii, A. A. (December 1979). "Spectrum of Relict Gravitational Radiation and the Early State of the Universe". Pisma Zh. Eksp. Teor. Fiz. (Soviet Journal of Experimental and Theoretical Physics Letters). 30: 719. Bibcode:1979ZhPmR..30..719S 24 Ibid Guth, Alan H. "Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems". Physical Review D. 23 (2): 347–356. Bibcode:1981PhRvD..23..347G. doi:10.1103/PhysRevD.23.347 25 ibid; Einstein, Albert (1916) 26 Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ch. 34, p. 916. ISBN 978-0-7167-0344-0. 27 Phillip D. Mannheim (April 2006). "Alternatives to Dark Matter and Dark Energy". Progress in Particle and Nuclear Physics. 56 (2). arXiv:astro-ph/0505266 Freely accessible. Bibcode:2006PrPNP. 56..340M. doi:10.1016/j.ppnp.2005.08.001 28 Trimble, V. (1987). "Existence and Nature of Dark Matter in the Universe". Annual Review of Astronomy and Astrophysics. 25: 425–472. Bibcode:1987ARA&A..25..425T. doi:10.1146/annurev.aa.25.090187.002233. 29 Kroupa, P.; et al. (2010). "Local-Group Tests of Dark-Matter Concordance Cosmology: Towards a New Paradigm for Structure Formation". Astronomy and Astrophysics. 523: 32–54. arXiv:1006.1647. Bibcode:2010A&A...523A..32K. doi:10.1051/0004-6361/201014892 30 The Dark Matter Group. "An Introduction to Dark Matter". Dark Matter Research. Sheffield, UK: University of Sheffield. Retrieved 7 January 2014 31 Conformal theory: New light on dark matter, dark energy, and dark galactic halos." (PDF) Robert K. Nesbet. IBM Almaden Research Center, 17 June 2014. 32 Ibid Phillip D. Mannheim (April 2006). 33 Ibid Nesbit (2014) 34 Hawking (1967) PhD Thesis Cambridge University Library 35 Heisenberg (1927), Heisenberg, W. (1927), "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", Zeitschrift für Physik (in German), 43 (3–4): 172–198, Bibcode:1927ZPhy...43..172H, doi:10.1007/BF01397280.. Annotated pre-publication proof sheet of Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, March 21, 1927 36 Dirac, P. A. M. (1939). "A new notation for quantum mechanics". Mathematical Proceedings of the Cambridge Philosophical Society. 35 (3): 416–418. Bibcode:1939PCPS...35..416D. doi:10.1017/S0305004100021162.. Also see his standard text, The Principles of Quantum Mechanics, IV edition, Clarendon Press (1958), ISBN 9780198520115 37 J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Chap. VI, pg. 417 (Princeton University Press, Princeton, 1955).

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Hawking, Stephen W. (1975). "Particle creation by black holes". Communications in Mathematical Physics. 43 (3): 199–220. Bibcode:1975CMaPh..43..199H. doi:10.1007/BF02345020. 39 Peebles, P. J. E.; Ratra, Bharat (2003). "The cosmological constant and dark energy". Reviews of Modern Physics. 75 (2): 559–606. arXiv:astro-ph/0207347  . Bibcode:2003RvMP...75..559P. doi:10.1103/RevModPhys.75.559. 40 Frieman, Joshua A.; Turner, Michael S.; Huterer, Dragan (2008-01-01). "Dark Energy and the Accelerating Universe". Annual Review of Astronomy and Astrophysics. 46 (1): 385–432. arXiv:0803.0982  . Bibcode:2008ARA&A..46..385F. doi:10.1146/annurev.astro.46.060407.145243. 41 Guth, Alan H. (1997). The Inflationary Universe: The Quest for a New Theory of Cosmic Origins. Basic Books. pp. 233–234. ISBN 0201328402. 42 Friedman, A: Über die Krümmung des Raumes, Z. Phys. 10 (1922), 377–386. (English translation in: Gen. Rel. Grav. 31 (1999), 1991–2000.) 43 Riess, A.; et al. (September 1998). "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant". The Astronomical Journal. 116 (3): 1009–1038. arXiv:astro-ph/9805201  . Bibcode:1998AJ....116.1009R. doi:10.1086/300499 44 Perlmutter, S.; et al. (June 1999). "Measurements of Omega and Lambda from 42 High-Redshift Supernovae". The Astrophysical Journal. 517 (2): 565–586. arXiv:astro-ph/9812133  . Bibcode:1999ApJ...517..565P. doi:10.1086/307221 45 The Value of the Cosmological Constant, 2011 Barrow, J. D., Shaw, D. J.; General Relativity and Gravitation 43, 2555-2560; arXiv:1105.3105 46 Aspects of the Cosmological “Coincidence Problem” (2011); H.E.S. Velten,∗ et al; Eur.Phys.J. C74 (2014) 11, 3160; arXiv:1410.2509 [astro-ph.CO 47 Urry, Meg (2008). The Mysteries of Dark Energy. Yale Science. Yale University 48 Courtois, Hélène M.; Pomarède, Daniel; Tully, R. Brent; Hoffman, Yehuda; Courtois, Denis (2013). "Cosmography of the Local Universe". The Astronomical Journal. 146 (3): 69. arXiv:1306.0091  . Bibcode:2013AJ....146...69C. doi:10.1088/0004-6256/146/3/69 49 Pomarede, Daniel; Hoffman, Yehuda; Courtois, Helene M.; Tully, R. Brent (2017). "The Cosmic V-Web". The Astrophysical Journal. 845: 55. arXiv:1706.03413  . doi:10.3847/1538-4357/aa7f78 50 Cottingham, W. N.; Greenwood, D. A. (2007). An introduction to the standard model of particle physics. Cambridge University Press. p. 1. ISBN 978-0-521-85249-4 51 M. S. Turner (1999). "Cosmological parameters". AIP Conference Proceedings. arXiv:astro-ph/9904051  . Bibcode:1999AIPC..478..113T. doi:10.1063/1.59381 52 G. Jungman; M. Kamionkowski & K. Griest (1996). "Supersymmetric dark matter". Physics Reports. 267: 195. arXiv:hep-ph/9506380  . Bibcode:1996PhR...267..195J. doi:10.1016/0370-1573(95)00058-5 53 Ibid; Gravitation, 1973 Charles W. Meisner, Kip S. Thorne, John Archibald Wheeler; ISBN 0-7167-0344-0 (pbk) 54 Ibid Gravitation, 1973 §42.3 Box 42.1 55 Ibid. §42.3 Figure 42.1 56 Slipher, V. M. (1913). "The Radial Velocity of the Andromeda Nebula". Lowell Observatory Bulletin. 1: 56–57. Bibcode:1913LowOB...2...56S 57 Hubble, E. (1929). "A Relation Between Distance and Radial Velocity Among Extra-Galactic Nebulae". 58 See Feynman, Leighton and Sands (1989) or any introductory undergraduate (and many high school) physics textbooks. See Taylor (1992) for a relativistic discussion. 59 Hawking, ibid; http://www.repository.cam.ac.uk/handle/1810/251038; 12 60 Silk, Joseph (2009). Horizons of Cosmology. Templeton Press. p. 208. 61 Ibid Gravitation, 1973 §27.6 62 Hawking, ibid; http://www.repository.cam.ac.uk/handle/1810/251038; 12 63 Hawking, S.W. Properties of an Expanding Universe; http://www.repository.cam.ac.uk/handle/1810/251038; 10 64 Hubble, Edwin (1929), "A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae" (PDF), Proc. Natl. Acad. Sci., 15 (3): 168–173, Bibcode:1929PNAS...15..168H, doi:10.1073/pnas.15.3.168, PMC 522427 Freely accessible, PMID 16577160 65 Ibid. 66 ibid. 67 J. Von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955)

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68

Ibid. J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Chap. VI, pg. 417 (Princeton University Press, Princeton, 1955) 69 Ibid. J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Chap. VI, pg. 417 (Princeton University Press, Princeton, 1955) 70 Manousakis, E. Found Phys (2006) 36: 795 71 F. Rieke and D. A. Baylor, Rev. Mod. Phys. 70, 1027 (1998). 72 K.-W. Yau, and D. A. Baylor, Ann. Rev. Neurosci. 12, 289 (1989). 73 E. Pöppel, Trends Cognit. Sci. 1, 56-61 (1997). 74 A. K. Engel, P. Konig, A.K. Kreiter, T. B. Schillen, W. Singer, Trends Neurosci. 15, 218 (1992). C. M. Gray. J. Comput. Neurosci. 1, 11 (1994). P. Fries, J.-H Schöder, P. R. Roelfsema, W. Singer, A. K. Engerl, J. Neurosci. 22, 3739 (2002). 75 ibid E. Pöppel, Trends Cognit. Sci. 1, 56-61 (1997). 76 TBS 77 TBS 78 TBS 79 Ibid. Gravitation, 1973 §1.3 pp. 19, 22 80 Einstein, Albert (1916). "The Foundation of the General Theory of Relativity". Annalen der Physik. 354 (7): 769. Bibcode:1916AnP...354..769E. doi:10.1002/andp.19163540702. Archived from the original (PDF) on 2012-02-06.

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