properties of an expanding universe based on an ...

11/21/2017

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: Mark S. Rogers

INTRODUCTION AND BACKGROUND: Skeleton Geometry1 Per Hawking's PhD thesis Introduction2, the expanding 'steady-state' model (supported by Edwin Hubble's initial observations) is assumed to have started from a point source Singularity and which presents the same appearance at all times. I propose that the universe began with a self-propagating point source 2-geometry Singularity which is continuing to propagate as a folded surface with curvature akin to a radially expanding homogeneously and isotropically infinite energy dense spherical 'shell'. The expansion of the 2-geometry surface possesses characteristics of an anterior surface (unmanifested) and a posterior surface having an event horizon. I propose that at the moment of Singularity, the 2-geometry instantly folded upon itself due to relativistic geometry and propagated outward leaving a vacuum on the posterior side of the expansion. I also postulate that the anterior expansion rate cannot be known as the event horizon prohibits any such observations. [In my model, posterior expansion can be measured from any theoretical point towards the expanding event horizon, thus the term 'posterior' meaning behind or remaining in the wake of the expansion vs. anterior meaning ahead of the event horizon and on the surface of the ‘shell’ 2-D singularity geometry.] So, although the 2-D expansion rate of the ‘shell’ cannot be known, the linear expansion rate on the posterior side can be observed and measured [We see this as the cosmological 'Red Shift' 1st noted by Edwin Hubble]. I postulate that this expansion results in characteristics including but not limited to vacuum, space, geometry, energy, density, and time. Within the posterior volume now enclosed by the expanding 'shell', this expansion rate has been observed and the characteristics are consistent with a universe described in the PhD Thesis by Hawking (ibid). In a review of Hawking's reference to Robertson's and Walker's metric3 for a model that is spatially homogeneous and isotropic {for edification, I include it herein as: ds2=dt2-R2 (t)[(dr2)/(1-Kr2 )] +r2 (dθ2+sinθ dφ2) for K = 0 or +/-14, we can apply this metric to equations that describe a dynamic 2-geometry model for an expanding universe. 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 1

Gravitation, 1973 Charles W. Meisner, Kip S. Thorne, John Archibald Wheeler; ISBN 0-7167-0344-0 (pbk) http://www.repository.cam.ac.uk/handle/1810/251038; 10 3 Ibid Gravitation, 1973 §27.6 4 http://www.repository.cam.ac.uk/handle/1810/251038; 12 2

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to substantiate a 2-geometry. 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 vertex.5 Let us allow for a sufficient number of vertices to exist 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. In Gravitation, we see how increasing the number of arbitrary paired lines about the vertex closely approximates a polyhedron built of triangles.6 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.7

(1)

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.8 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. In an expanding 2-dimensional surface, 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 (Eq 1). Eq. 1 shows that a 2-geometry Singularity structure under conditions of curvature can fold on itself to become a smooth continuous surface of a sphere.

5

ibid Gravitation, 1973 §42.3 Box 42.1 Ibid. §42.3 Figure 42.1 7 Ibid. 8 Ibid. 6

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ANALYSIS OF AN EXPANDING 2-D SINGULARITY 2-D Singularity and Expansion We begin by introducing Dirac notation to model concepts such as potential expansion, and operation of expansion which are necessary in order to discuss our proposal. First, the operation of expansion can be thought of as an operation which causes the state of potential expansion 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⟩. If we assume our Singularity to have a potential function for expansion then it follows that we can define the potential for Singularity expansion herein as our |ψi⟩. We can also agree that 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. 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 Pg 3 of 8

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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:

I = ⟨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 ∞ (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. To verify we have found an expansion independent of the initial state, we may start from the state |ψ0⟩ = ∑Mi=1 |x10 ⟩ 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⟩).

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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⟩



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.9 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⟩ 

(6)

(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 expansion. The result of the measurement 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 measurement 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:

9 J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Chap. VI, pg. 417 (Princeton University Press, Princeton, 1955)

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λ⟩ = ∑μ ψλ(μ)|μ)

(9)

ψλ(μ) = μ|λ).

(10)

To apply this expression to an event of expansion, let us suppose that the measurement of the expansion represented by the operator Ô transforms the state of potential expansion to a particular eigenstate |λ⟩. The result of the measurement is the corresponding eigenvalue λ. We do not assume that the next expansion event is in any way related to the previous expansion measurement 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 measurement 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. As eigenstates correspond to arbitrary expansion events, let us represent a body of arbitrary expansion operators Ɵ̂ on our 2-D singularity surface by:

∑i=1 ꝏ 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:

ƴ i⟩

= ∑ƴi ψƴi (μi)| μi)

ψƴi (μi) = μi|ƴi).

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(13) (14)

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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)

ψδi (μδi) = μδi|ƴδi).

(15) (16)

Discussion Of The Main Results 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. This is akin to a number line of real numbers that stretches from -ꝏ to +ꝏ where upon closer examination we find an infinite number of real numbers on an arbitrarily chosen interval between two real numbers on our number line. Thus, we argued that 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 expansion. On an expanding 2-D Singularity surface, we showed that arbitrary states of potential expansion can have unique representations in Hilbert space and that the unique states of potential expansion can have unique eigenvectors that consist of unique expansion basis vectors which are composed of eigenvalues that are the unique deficit angles.

In the next section, we hope to argue that our expanding 2-D Singularity surface is a modality for an expanding universe consistent with conclusion shown by Einstein10, Hawking11, Wheeler12, and others.

10

TBS TBS 12 TBS 11

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TBS - WIP Let us consider the potential for expansion in a set of ordered pairs in our 2-D Singularity surface to be { Substituting N = ꝏ into Eq 4

Gravitational Theory 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 ‘red- shift' 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.13

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

13 Einstein; TBS

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