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Nutshell-2019
19th April 2020
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"Plato is my friend; Aristotle is my friend, but my greatest friend is truth." - Isaac Newton "Un homme heureux est trop content du présent pour trop se soucier de l'avenir. (A happy man is too satisfied with the present to dwell too much on the future.)" - Albert Einstein "We should consider the laws of Nature as continually changing with the epoch." - Paul Dirac "Is the purpose of theoretical physics to be no more than a cataloging of all the things that can happen when particles interact with each other and separate? Or is it to be an understanding at a deeper level in which there are things that are not directly observable (as the underlying quantized fields are) but in terms of which we shall have a more fundamental understanding? " - Julian S. Schwinger “In other words, the laws of Newton could be stated not in the form F=ma but in the form: the average kinetic energy less the average potential energy is as little as possible for the path of an object going from one point to another." - Richard P. Feynman "Reading book after book the whole world died, and none ever became learned! But understanding the root matter is what made them gain the knowledge! " - Kabir Granthavali, XXXIII.3, Translated by Charlotte Vaudeville ***
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Earlier we met with Aku the Amoeba and his commute in the universe known as “Blip” and as it happened Blip was infested. There was this quantity which did not change as Bliponians zipped through various Möbius and Lorentz Transformations, a quantity which measured this magnificent infestation, called entropy.
The action-gauge
which was supposed to provide
information about Blip, was also
compromised with entropy. Which
meant Bliponians, were traveling blissfully unaware
on a time-line
with only one way forward, straight into
a black-hole. Now Bliponians
were not only unaware of entropy but
they also did not know that the
entropy was continuously increasing
with the progression of time.
So if an electron in Blip, was
shaken more than 1372 times
before it spat out a photon.
Then few millions years down
the time-line same electron needed to
be shaken more than 1382
times before a photon popped-out. The fundamental e-γ interaction, represented the measurement-tools the observers were using for their measurements in Blip. The entropy inherent in the system not only limited the effectiveness of these measurement-tools, but also ensured that these measurement-tools became useless with the progression of time. The situation was as follows:
The black-hole in
j-space, defined the state where the
information about the initial state
was completely lost, and maximum
granularity in measurements was
achieved (a.k.a. quantum
computer). This was where the
concept of life-force ceased to
exist, however an observer in
j-space would never be able to
measure the exact structure of the
black-hole. In essence, except
for the existence of life (or
equivalently the initial state), the
whole universe was sitting inside a
black-hole. Thus any measurement made in any manifold in the Bliponian universe had to follow the topological constraint defined by the relationship above. The dimensions of the constant jML, were Mass × Length. No condition was imposed upon the nature of the mass and the length, except for the fact that unlike classical region, they were not independent of each other in the topological space. Similar constraint was also placed on the fundamental constant h and c, such that in the topological space h and c were not independent of each other. The relationship shown above, could be written in terms of Compton's Wavelngth as: where mo was the rest mass. We could apply the criterion that the time dependence must be eliminated, to get the equivalent topological representation of a Black-hole (BH) as well. The Schwarszchild metric in General Theory of Relativity, was given as: Here the coefficient of dt2 term must be zero. From there the expression for the mass of the BH in topological space, could be derived as: ,
which essentially
is the Planck mass. The
estimated mass of a BH was of the
order of solar mass,
approximately 2 × 1030 kg.
Yet in the topological space, this
mass was equivalent to Planck
mass, 2.18 × 10-8 kg.
The difference was
truly extraordinary. We can also think of this as duality between the measurements made by ObsM and Obsc. So the question arose, what did the quantity known as mass really represent? Was the value of the measured mass an intrinsic property or was it observer dependent? For a given measurement in j-space, Planck-mass mP, Zero-entropy Mass mZE, and in this case Solar Mass MBH in Kruskal-Szekeres coordinates, were shown below: Blip existed in the Region-I, so did MBH. The solar mass MBH was measured by ObsM in the Region-I. The Planck mass mP was measured in the Region-II and it was measured by Obsc. The zero-entropy mass mZE corresponded to the mass measured by Obsi. It was finite but unmeasurable by either ObsM or Obsc. (The continuity was defined as the succession of zero-entropy measurements.) If the radius of Blip was estimated as 4.4×1026 meters, then the definition of the constant jML could be used to estimate the mass of the Blob underlying Blip as 10-67 kg, an incomprehensibly small number. For comparison the mass of the electron, a Blip entity with no known internal structure, was 9.1×10-31 kg. An ecosystem of δj-functions: Consider the internal structure of Blip, without planets, stars, galaxies, BHs etc. In j-space there was no perfect vacuum as a perfect "null" could not be measured. For any arbitrary time instant, the fabric in j-space consisted of j-pixels which were seamed together by the Lorentz Invariance i.e the neighbouring j-pixels were bound by Lorentz Invariance. And since Lorentz Transformation and Möbius Transformation were equivalent to each other, at any given instant on Blip time-line, the structures measured in the neighbouring pixels could be transformed into each other using Möbius transformations. Single Element Matrices in j-space: So how can Aku and his team, think about the field-equation shown below, describing the space-time fabric of Blip, in the topological space? 1. Λ is invertible iff it is non-singular. 2. More importantly Λ is always non-singular in j-space. The argument is straight forward. Λ is singular iff Λ = [0]. However in j-space [0] is equivalent to [0j] which always has a finite value corresponding to the least energy surface and hence Λ is non-singular and invertible both. 3. Thus the condition that any metric tensor in GR has to be invertible, derives from the requirements of the underlying topological space, and it also prohibits the existence of pure vacuum in j-space since Λ is always finite. However the condition of invertibility also prohibits the existence of a pure solid as well. It can be stated as that the condition of "totality" in either direction is prohibited in Blip, per the requirements of the topological space underlying Blip. 4. Furthermore since Λ is invertible and non-singular, we can derive the condition for continuity as, B = Λ-1AΛ. Here A and B are also single-element matrices and hence by definition represent zero-entropy measurements in j-space, and thus continuity i.e. B and A are seamlessly connected via Λ. (Note that Λ is derived from gμν and it is not a scalar in the topological space. We are merely counting in the topological space using single-element matrices and Λ.) 5. In exterior Region-I, Λ can be represented as eiH, where H is a Hermitian matrix. Since Λ by definition is non-singular, it is not a zero-trace matrix describing a convex surface. Whereas no such restriction is placed on H, which converges to a zero-trace matrix under the relativistic limits. Thus eventual metric corresponding to the topological structure is always finite, though it can be represented in terms of zero-trace matrices in a measurement space corresponding to a lower information content. 6. The element in single element matrix, is not a number or a scalar per say. The definition of the element is based on the zero-entropy requirement of the topological space. The element can be represented as following: Here the
element is shown just as a blob
with undetermined characteristics,
rather than a number. The
definition of the
element is purely a discrete
measurement construct. It
represents the limitation of the
observer making the
measurements. The
element
itself can have its own eigenvalue
spectrum corresponding to a higher
information space, not accessible to
either Obsc or ObsM.
Another important issue is the definition of the inverse of the element. The entities "1" and the operation of inversion, have very different meanings in the topological space, unlike the standard definitions used in exterior region-I of KS representation discussed earlier. More importantly we must understand, why do we need to perform the inversion operation on the blob, to begin with? What does this operation signify? Continuity in j-space: Finally, the concept of continuity exists only in the context of topological space, which actually seems like a "particle" property1. Traveling on a straight time-line in the Region-I is more analogous to moving in circles but not continuity, and it seems like a "wave" property. However in the core of all this, is the issue of the "measurement" using a fundamental interaction. The "measurement" of the topological structure underlying Blip, by using the tools based on a manifold comprising of fundamental interactions such as e-γ and others. We will discuss single element matrices and the corresponding structures later on. We have already defined the challenge for ourselves. We will be discussing the Theory of Special Relativity in the Information Space of an infinite source first. Then we need to understand the Theory of General Relativity and required Tensor structures. Penrose diagrams are quite important. Anharmonic coordinates along with the constructibility condition, represent a powerful tool to describe an interaction. However all said and done, we must describe the Bliping problem in sufficient details first. That is all in a Nutshell-2019! *** |
Previous Blogs: Stitching the Measurement Space - III Stitching the Measurement Space - II Stitching the Measurement Space - I Mass Length & Topology Chiral Symmetry
Sigma-z and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous z-Axis Majorana ZFC Axioms Set Theory Nutshell-2014 Knots in j-Space Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions b-Field & 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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