|Möbius and Lorentz Transformations
Equivalence in j-space - I
30th July 2018
"The complex mapping that correspond to Lorentz transformations are the Möbius transformations! Conversely, every Möbius transformation of C yields a unique Lorentz transformation of space-time.
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 space-time 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 j-space in C. Both Möbius and Lorentz transformations shall play crucial roles in forming various structures in j-space.___________________
But first let us set up the problem of measurement. A finite-capacity observer (v/c << 1) ObsM, is in the state of measurement of an infinite source. We call this observer a macroscopic observer, for example human beings. In the j-space, the infinite source is defined as "a point containing infinite information" within itself 1.
Since in reality, all the information within the-point can never be precisely measured by ObsM, we also called the infinite information source as "indeterminate or i-space". The observer ObsM, can precisely measure only an infinitesimal portion of "indeterminate or i-space", called δi. The subscript i refers to the information measured from the i-space.
As the observer ObsM is in the state of measurement, δi essentially represents the universe (∞j) the macroscopic observer exists in, also called the j-space. We represent this rather annoying revelation as,
∞j = δi.
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 δi, can be represented by the unit-point sphere SU. In the following figure SU 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 Obs1 and Obs2 are (t1, x1, y1, z1) and (t2, x2, y2, z2) 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 space-time interval Δs2 between two events as measured by either of them, will be the same, i.e.
s2 = (c2Δt12 - Δx12 - Δy12 - Δz12) = (c2Δt22 - Δx22 - Δy22 - Δz22).
Thus if the observers are able to measure the unit-point sphere completely, albeit their descriptions of the events and how they occur occur may be different, their measurements must be Lorentz Invariant i.e. Δs2does not change for either of them2. 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 j-space 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 = 0j>. 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 ObsM but that of ObsC instead. (ObsM does not have the memory of the initial state, ObsC does.)
In j-space the condition of Lorentz Invariance is enforced by the measurements made by the observer Obsc (v/c =1), the maximum efficiency observer in q=3 state. We can also say that the details of the universe as measured by Obs1, will be linearly correlated to the details of the universe as measured by Obs2, 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 j-space. 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 shape3. This is a very important property as it uniquely allows the translation of the information without loss, from inside the unit-point 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 real-life.)
One but can not help notice the similarities with wave-theory 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 electron-positron 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 space-time continuum Δs2= gik dxi dxk, the invariance of Δs2 is also stated as the imposition of the metrical connexion up on the space-time continuum. The metric gik 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.
Sigma-z and I
Rationale behind Irrational Numbers
The Ubiquitous z-Axis
Knots in j-Space
Cauchy and Gaussian Distributions
Discrete Space, b-Field & Lower Mass Bound
The Cat in Box
The Initial State and Symmetries
Discrete Measurement Space
The Frog in Well
Visual Complex Analysis
The Einstein Theory of Relativity
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