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The General Relevance of the Modified Cosmological Model Jonathan W. Tooker December 25, 2017 Contents I Introduction ......................................... 1 I.1 An Abstract Psychological Dimension . 1 I.2 The Dual Tangent Space . 6 I.3 Feynman, Functions, and Functionals . 14 II General Relevance with Emphasis on Gravitation . 24 II.1 Relevant Aspects of Classical Physics . 25 II.2 General Relativity . 40 II.3 An Entropic Application . 52 II.4 Complex Coordinates . 53 II.5 What is' ^? .......................................... 57 II.6 Twistors and Spinors . 61 II.7 Dyads and Quaternions . 73 II.8 Unification . 80 III Maximal Symmetry .................................... 89 III.1 The MCM Hypothesis . 89 III.2 Historical Context . 93 III.3 A Few Miscellanea . 98 III.4 Problems One and Two . 102 III.5 What is a Multiplex? . 105 III.6 Problems Three and Four . 106 III.7 Maximally Symmetric Spacetime . 110 III.8 Toward Geodesics . 118 III.9 Dark Energy and Expanding Space . 125 III.10 Advanced and Retarded Potentials . 126 IV Computation and Analysis in Quantum Cosmology . 130 IV.1 The Modified Cosmological Model . 131 IV.2 Tipler Sinusoids . 167 IV.3 MCM Quantum Mechanics . 173 IV.4 Conformalism and Infinity . 198 IV.5 Covering Spaces . 218 IV.6 The Double Slit Experiment . 228 IV.7 Fundamentals of Hypercomplex Analysis . 238 IV.8 Two Stringy Universes in the Information Current . 262 V Death to Detractors ...................................268 Appendices 269 A Synopsis of Historical Development 270 Bibliography 277 1 The stone the builders rejected has become the cornerstone. { Psalm 118:22 I Introduction In lieu of an abstract each chapter in this book will have a description of its contents. This book is focused on recapping, consolidating, streamlining, and annotating previous work related to gravitation and non-relativistic quantum theory while adding a few new insights when they are modest. Throughout this book the reader's familiarity with the modified cosmological model is assumed but not strictly required. The focus of the first section in this chapter is a review of geometry. Section two gives a preliminary overview of an algorithm that will violate conservation of information. In section three we propose to modify Feynman's application of the action principle by replacing the least action complex field trajectories with maximum action hypercomplex field trajectories that still satisfy the action principle. I.1 An Abstract Psychological Dimension It is shocking that after this many years of work on the theory of infinite complexity that the associated material calculated and referred to here is not already well known with the entire field of all possible linear nuance being mapped out to the nth degree. It is surprising that there is no Wikipedia article regarding the modified cosmological model (MCM) or the theory of infinite complexity (TOIC) that spells out all of the trivially derived properties. To that end, consider a cube spanned byx ^,y ^, andz ^. The slices of constant z are the subspaces spanned byx ^ andy ^ at each value of z. Every curve that can be constructed usingx ^ and y^ will be confined to some slice of z. Any curve leaving the slice would have a component in thez ^ direction. Likewise any curve constructed from justx ^ andy ^ will have its tangent vectors confined to that single slice of constant z. The curve's cotangent space is the first place we could possibly come across vectors with a non-vanishingz ^ component. We state these obvious truths because the MCM describes de Sitter (dS) and Anti-de Sitter (AdS) spacetimes as slices of a 5D cube and we want to show the exceptional behavior of our flat universe when it sews together two 5D spaces but is not itself a slice of any 5D space. Now consider flat empty 5-spaces Σ± where general relativity in the absence of 5D matter- energy leads to the desired dynamics in the 4D slices through the Kaluza-Klein metric 2 Figure 1: The region between two adjacent moments of psychological time: H1 and H2. The arrangement immediately suggests a gravitational pilot wave formulation as the path to evolve through the discontinuity of the as-yet-undescribed region inside the null interval between Ω1 and @2 but we will introduce another simpler formulation in this book. We will introduce new χA coordinates ? to accommodate this representation wherein xµ 2H are moved away from the center of the MCM unit cell where we have depicted them in previous work. χ5 is the horizontal direction across this figure. This figure uses the values Φ2, Φ, and 1 to demonstrate Σ? but there are many such arrangements. 2 2 ! gαβ + φ AαAβ φ Aα ΣAB = : (1.1) 2 2 φβ φ In this book we will use the Greek letter χ for the 5D coordinates where we have used ξ previously. Where Latin indices A have previously run from 0 to 4, here they will run from 1 to 5 so ξ4 !χ5. We will add a layer of complexity when we take µ, ν 2f0; 1; 2; 3g in the usual way but then add a subtle convention for α; β 2f1; 2; 3; 4g. In 5D we have A; B 2f1; 2; 3; 4; 5g _ ± A and α; β 2f1; 2; 3; 4g orα; _ β 2f2; 3; 4; 5g. Taking the coordinates of Σ as χ± we will call the ± bulk metrics ΣAB and they will have the form of equation (1.1). Curves in the flat slices of 5 constant χ± can never have tangent vectors that point to the left or right in the cosmological unit cell. Figure 1 shows that cell. The slices @ and Ω are flat slices of χ5 but they appear curved in this figure to demonstrate the curvature associated with the embedded metric of 3 µ α the de Sitter coordinates x± 6=χ±. The oft-lamented \cylinder condition" that the MCM both embodies and motivates from µ first principles [1, 2, 3] says that physics in the 4D worldsheets spanned by χ± can never depend on the fifth coordinate. This can be accomplished via a generalized disallowance 5 of the appearance of χ± in any equations of motion but we can accomplish the same thing 5 by taking our 4D spacetimes as surfaces of constant χ± in the 5D bulk [3]. The ordinary limitation of the cylinder condition on physics is that position and momentum measured in xµ can never depend on x4. However, that doesn't say anything about the abstract coordinates A A A fχ+; χ?; χ−g or vice versa. Here we begin to develop the complex behavior that can be derived by modeling our universe of xµ at the intersection of two 5D spaces Σ±. This is a key point to notice: observables will always be defined on xµ which can, in principle depend on all of the χA coordinates. This contrasts the normal application of Kaluza{Klein theory which says xµ cannot depend on x4. Therefore, even at this early stage, it is apparent that the MCM is very different from the standard cosmological model and other Kaluza{Klein models. One well known issue with standard Kaluza{Klein theory is that the field equations indicate that the electromagnetic field strength tensor must always vanish with respect to 4D general A A A relativity. By adding the 15 chirological coordinates fχ+; χ?; χ−g we have a lot of room to develop novel workarounds. For instance, if the Kaluza{Klein requirement for vanishing electromagnetic strength tensors applies to the chirological coordinates then that puts only µ µ µ a loose constraint on what we do with the x , x?, and x± coordinates. Let χ5 be a non-relativistic psychological dimension with identical topological flatness. 5 5 The identical topological flatness of χ± does not hold for χ? which can have an arbitrary non-linear curvature with tangent vectors pointing anywhere because it has no width in the ? ± path from H1 to H2, as in figure 1. Σ exists only to sew Σ together with a single point so we are not concerned with the overall curvature there. There is no constrained object anywhere in the vector bundle of Σ? so everything about that bundle is introduced as a new MCM degree of freedom. The only constraint on Σ? is that it has at least one point where we can construct a Lorentz frame and then use that point to ensure smooth transport of a Lorentz frame from H1 to H2. The 4D slices of flat 5-space are flat but @ and Ω, themselves slices, are curved and what's more: the only flat space we do have, H, isn't even a slice of a 5-space because Σ± do not contain their boundary at χ5 = 0 which specifies the location of H [3]. H is the unincluded boundary of two 5D half spaces. How can we get a curved slice out of a flat space? These new degrees of freedom beyond H will be helpful. The addition of only one new degree of MCM freedom to go through larger infinity in the hyperreal number system ∗R (via Φ^ n !Φ^ n+1) leads to two new degrees of freedom: the two dimensions of C become hyperreal and hyperimaginary. We will name the system that contains hyperreal and hyperimaginary numbers the hypercomplex number system1 ?C. We point to hyperimaginarity as the reason for the fourth ontological vector 2^ which allows us to use f^i; Φ^; 2^; π^g as a basis for general relativity (or rather we might choose to call the fourth one ^i because it more precisely corresponds to hyperimaginarity.) Our initial desire to add a single degree of freedom in a longitudinal mode along Φ^ showed that fπ;^ Φ^;^ig was incomplete [4, 5].

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