

Möbius and Lorentz Transformations Equivalence in jspace  I 30^{th} July 2018

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"The complex mapping that correspond to Lorentz transformations are the Möbius transformations! Conversely, every Möbius transformation of C yields a unique Lorentz transformation of spacetime. Even among professional physicists, this miracle is not as well known as it should be."  Tristan Needham in Visual Complex Analysis. "What we hope is at some point we will see something in the galactic centre that we can't explain with Einstein's theory  that would be really, really exciting. Because then we could go back to the drawing board and come up with something better." 
We will discuss the equivalence between Lorentz transformation in Special Theory of Relativity and Möbius transformation in Complex Space C. The Lorentz transformation ensures that two observers moving with respect to each other with a constant velocity, preserve the spacetime interval between two events. However a physical picture seems to be missing. What about respective geometric shapes? Would a sphere measured by an observer would also be measured as a sphere by another in a relative frame, or as a cube. How about the relationships, temporal and spatial both, between geometric shapes? In next few blogs we will be discussing the characterization of jspace in C. Both Möbius and Lorentz transformations shall play crucial roles in forming various structures in jspace.
___________________But first let us set up the problem of measurement. A finitecapacity observer (v/c << 1) Obs_{M}, is in the state of measurement of an infinite source. We call this observer a macroscopic observer, for example human beings. In the jspace, the infinite source is defined as "a point containing infinite information" within itself ^{1}. Since in reality, all the information within thepoint can never be precisely measured by Obs_{M}, we also called the infinite information source as "indeterminate or ispace". The observer Obs_{M}, can precisely measure only an infinitesimal portion of "indeterminate or ispace", called δ_{i}. The subscript i refers to the information measured from the ispace. As the observer Obs_{M }is in the state of measurement, δ_{i} essentially represents the universe (∞j) the macroscopic observer exists in, also called the jspace. We represent this rather annoying revelation as, ∞j = δ_{i}. A bummer isn't it? Here we are, fantasizing about the supremacy over "others" by any means possible, yet what we know including life, may not even be an infinitesimal of what we really do not know. An amoeba in q=2 space is probably more capable than the whole universe combined in q=3 space. For macroscopic observers, the information ∞_{j} or δ_{i}, can be represented by the unitpoint sphere S_{U}. In the following figure S_{U} is being being measured by two observers who are in relative motion with constant velocity, with respect to each other. The observers would be in disagreement with respect to the time and the space coordinates. Assume that the time and the space coordinates for Obs_{1} and Obs_{2} are (t_{1}, x_{1}, y_{1}, z_{1}) and (t_{2}, x_{2}, y_{2}, z_{2}) respectively. However the observers are restricted by the requirement that maximum velocity possible is c, which is the speed of light.
The measurements performed by both observers can be linearly mapped into each other using Lorentz transformations. The consequence, is that they will have to agree that the spacetime interval Δs^{2} between two events as measured by either of them, will be the same, i.e. s^{2} = (c^{2}Δt_{1}^{2}  Δx_{1}^{2}  Δy_{1}^{2}  Δz_{1}^{2}) = (c^{2}Δt_{2}^{2}  Δx_{2}^{2}  Δy_{2}^{2}  Δz_{2}^{2}). Thus if the observers are able to measure the unitpoint sphere completely, albeit their descriptions of the events and how they occur occur may be different, their measurements must be Lorentz Invariant i.e. Δs^{2}does not change for either of them^{2}. The Lorentz Invariance is required due to the fact that time and space are not linearly independent of each other. The information gathered from the same space point at different time instants is not independent of each other, unless assumed otherwise as in Newtonian Mechanics. Similarly the information gathered at two different space points at the same time instant is not independent of each other, unless assumed otherwise as in Newtonian Mechanics. (In jspace the terms "events" and "information gathered from measurements" are analogous.) It is also worthwhile to note that the independence of the space and time coordinates occurs only when there is no memory of the initial state <t = 0_{j}>. In this case the structures are represented by the uniform distribution and the macroscopic observer is allowed to start and stop his or her measurement clock as he or she wills. The uniform distribution is not a stable distribution. As the memory of the initial state is brought in to the picture, we move towards stable distributions such as Gaussian, Cauchy and Levy. In this case, the structures are not governed by the measurement clock of Obs_{M} but that of Obs_{C} instead. (Obs_{M} does not have the memory of the initial state, Obs_{C} does.) In jspace the condition of Lorentz Invariance is enforced by the measurements made by the observer Obs_{c} (v/c =1), the maximum efficiency observer in q=3 state. We can also say that the details of the universe as measured by Obs_{1}, will be linearly correlated to the details of the universe as measured by Obs_{2}, by Lorentz transformations. Please note that the linear correlation between the details of the universe in complex plane, does not imply linear independence of time and space coordinates as in Newtonian mechanics. Now imagine that both observers are comparing observations made at "the same instant" in the jspace. What will be the relationship between their measurements? One thing for sure, is that the measurements must be linearly correlated as the Lorentz Invariance must still hold. Thus at a given instant if two observers either stationary or moving with a constant velocity with respect to each other, make measurements then, i) The Lorentz Invariance must be satisfied, and ii) The measurements at that instant are correlated by the Möbius Transformation in C. Why Möbius Transformation? Because Möbius Transformation preserves forms in Complex space C. For example if an observer measures a circle in the interior, then it is mapped using Möbius Transformation, as a circle in Λ_{∞ }region. Furthermore Möbius Transformation is the unique transformation in C, which preserves the structure shape^{3}. This is a very important property as it uniquely allows the translation of the information without loss, from inside the unitpoint sphere where it is measured, into the Λ_{∞ }region where the macroscopic observers can visualize it. (Note: We will have to show how structures in C, transform into physical structures we observe and measure in reallife.) One but can not help notice the similarities with wavetheory and matrix formulations of quantum mechanics. The matrix formulation is based on the description of a system at a given instant which also includes the spin effect. The wave theory does not include spin in its formulation and hence the concept of the electronpositron pair, does not exist. The time in quantum wave theory is more of classical nature as it is used in conventional wave theory. This is turning out to be a rather fascinating discussion. We will continue to explore this extraordinary result further. 1. The exact nature of the information is not known. The information could be related to biological, chemical, physical or mathematical structures or a combination of them. Only conditions we place are, that the observer making measurements can report only what it could measure scientifically, and that the observer will remain in the state of measurement until the PE1 measurement is complete. 2. In spacetime continuum Δs^{2}= g_{ik} dx^{i} dx^{k}, the invariance of Δs^{2 }is also stated as the imposition of the metrical connexion up on the spacetime continuum. The metric g_{ik} represents scalar density. In conventional (t, x, y, z) metric, the metrical connexion translates into Lorentz invariance. 3. For details please refer to Needham's Visual Complex Analysis. ***

Previous Blogs: Chiral Symmetry
Sigmaz and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous zAxis Majorana ZFC Axioms Set Theory Nutshell2014 Knots in jSpace Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, bField & Lower Mass Bound Incompleteness II The Supersymmetry The Cat in Box The Initial State and Symmetries Incompleteness I Discrete Measurement Space The Frog in Well Visual Complex Analysis The Einstein Theory of Relativity *** 

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