

Mass, Length, and Topology
19^{th} May 2019

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"There are at present fundamental problems in theoretical physics awaiting solution, e.g., the relativistic formulation of quantum mechanics and the nature of atomic nuclei (to be followed by more difficult ones such as the problem of life), the solution of which problems will presumably require a more drastic revision of our fundamental concepts than any that have gone before."  P.A.M. Dirac; Proc. Roy. Soc. A 133, 60 1931. " In any case, models must, of course, be constructed in accordance with acceptable physical theory since the values which distinguish the cosmological speculations of the scientist from those of the crank arise from the attempt of the former to make his work logical and coherent with the rest of physics."  Richard Chance Tolman; Relativity, Thermodynamics, and Cosmology. "Neither space nor time has any existence outside the system of evolving relationships that comprises the universe. Physicists refer to this feature of general relativity as background independence."  Lee Smolin; Three Roads to Quantum Gravity. 
Previously we introduced a constant j_{ML} with dimensions of MassLength, for the topology in the discrete measurement space, defined as:
One would expect a dimensionless constant for a true topological space. However we are developing a description for the discrete measurement space or jspace, in which the topology is defined as the entropyfree space. The entropyfree space implies that time will not be a factor. Thus the constant j_{ML}, is timeindependent. (Topology represents the path connectivity in measurement space, we will discuss it in a later blog.)
where m_{o} is the rest mass for elementary particles, the value of j_{ML} should come out to be the same for every elementary particle. The following table shows the calculated value of j_{ML} for some of the elementary particles^{1}:
No surprises here, as λ_{Compton}'s are calculated values to begin with. It is just a sanity check confirming that in the relativistic region, which represents the underlying topological structure for a discrete measurement space, the value of the timeindependent constant j_{ML}, does not change. Here we will like to point out that, it is a standard practice in relativistic physics to assume h = 1 and c = 1. However in a discrete measurement space we place the ratio h/c equal to one, instead of h = 1 and c = 1 individually ^{2}. (h÷c = 1 leads to Mass × Length = 1 or M = Length^{1}, the length in this case would represent λ_{Compton}.) Now the fun part! Let us consider next a relativistic structure observed at the cosmic scale, a black hole. The Schwarszchild metric in General Theory of Relativity, is given as:
We move into the topological space by making the metric g_{Sch} time independent, i.e. we equate the coefficient of dt^{2} to zero^{3}. This gives us, Schwarszchild radius as, R_{Sch} is the radius defining the eventhorizon of a black hole, and M_{BH} is the mass of the black hole. We can define the corresponding wavelength as,
Subsequently using the criterion that the constant j_{ML}, representing the inherent topology of a given measurement space, does not change for all the structures in jspace, we can write, Substituting for λ_{Sch} in above expression, we obtain the expression for Planck mass, . The estimated mass of a blackhole is of the order of solar mass, approximately 2 × 10^{30} kg. In topological space, this mass is equivalent to Planck mass, 2.18 × 10^{8} kg. The difference is truly extraordinary. So what does the mass really represent? Is the value of the measured mass an intrinsic property or is it observer dependent? Let us first review the information in hand.
1. Planck mass corresponding to R_{Sch} is measured for the least energy surface surrounding the topological structure. The actual black hole mass in units of solar mass is measured in the metric space or the reference frame of a macroscopic observer Obs_{M}, where the least energy surface acts as the definition of the origin. The actual information content of both measurements remains the same, yet mass measured, differ by many orders of magnitude. 2. Aku and his Bubble represent the macroscopic observer Obs_{M}, who measures M_{BH} in units of solar mass. Let us represent R_{Sch} in KruskalSzekeres coordinates as shown below. In this representation, we do not have to worry about the geometrical singularity at the origin ^{3}. The Aku's measurements estimating M_{BH} are in the exterior regionI. The Planck mass M_{P} lies in the interior regionII. So how do we reconcile two very different values for the measurements of mass of same topological object?
If we recall, R_{Sch} represents the value for black hole dimensions, below which the inner mechanism of the black hole become important. This is also the region where Euclidean geometry used to describe the regionI, is no longer applicable. Aku or Obs_{M} has fairly limited capability (v << c). Assuming that Aku proceeds into the black hole to make required measurements, and if his progress is monitored by another observer in the regionI, per this observer it will take Aku forever to enter the black hole. Thus it is impossible for an observer in the regionI who is characterizing the black hole in terms of physical quantities such as temperature, mass, angular momentum, and lifetime etc., to precisely measure the insides of the black hole. The entropy of black hole measurement is very high, due to limitations of the electronphoton interaction (q = 3 space) based measurement apparatus available to Aku. Even if Aku was inside R_{Sch}, the amount of information which needs to be measured is enormous and consequently in all likelihood Aku will continue to move towards the black hole without successfully completing the required measurements. Let us review the relationship between G and M_{BH , }corresponding to the topology of the interior regionII, . In order to move to the topology in the exterior regionI, we can replace M_{BH} with the ZeroEntropy mass m_{ZE}_{ }, measured by the maximum efficiency observer Obs_{c}. Recall that the zeroentropy observer means the condition ds^{2} = 0_{j}, where ds is the spacetime interval given as ds^{2} = g_{μν }ds^{μ}ds^{ν}, and therefore mass m_{ZE}_{ }_{ }corresponds to the condition for the spacetime interval equal to zero (0_{j}). The relationship becomes,
We note that on the right hand side we have universal constants h and c, along with zeroentropy mass m_{ZE}_{ }_{ }. Since the mass m_{ZE}_{ }_{ } is measured by Obs_{c}, it will also act as an universal constant for Aku's (equivalently Obs_{M}'s) measurements in exterior regionI. Therefore in exterior regionI, G is expressed in terms constants h, c, and m_{ZE}_{ }, each of which is a universal constant for the exterior regionI, and hence G itself is a universal constant for the exterior regionI (only), denoted by G_{I}. Note that G_{I} is not G_{Newton}, the known universal constant of gravitation. (Note: KnotTheory describes exterior region1.)
Similarly to derive the relationship for G in the regionII, we replace m_{ZE} with m_{p} , and use the relationship G_{II} = hc÷(2πm_{p}^{2}), where m_{p} is Planck mass. At this point we can not be sure that m_{ZE}_{ }_{ }and m_{p} are equal, unless they can be precisely measured. (More importantly the value of G in the regionIV G_{IV} , can not assumed to be equal to G_{II} .)
What if there was another observer, Obs_{i} from higher information space^{4}, whose measurement instruments are much more advanced? Obs_{i} is able to measure all the information inside the black hole in one measurement, i.e. Obs_{i} is able to perform a zeroentropy measurement same as Obs_{c}, plus Obs_{i} has higher capabilities. Then we can ask, what would be the nature of the black hole for Obs_{i}, a shallowwell (just another inconvenient pothole) or more formidable infinitepotential well (actual black hole)? In other words, would a black hole in q = 3 information space, exist as a black hole in q = 2 or q = 1 information spaces? One thing for sure, the observer from q = 3 information space, definitely does not want to know about black holes in q = 2 or q = 1 information spaces. ___________________ 1. Based on the values for Compton wavelengths from National Institute of Standards and Technology (NIST). The units are in MKS. We will also like to point out that by making metric g_{Sch} time independent, the coefficient of the dr^{2} term, (~c^{2}/0_{j}), becomes extremely large. This term in not measurable. Subsequently the effect of variation in a time independent metric corresponding to the topological space δ_{i}, will show up as variations in dθ^{2} and dφ^{2}, i.e. rotational components only, with a massive singularity accompanying dr^{2} term in Aku's measurement space. No surprises here as we already know that internal symmetries do show up as various angular momenta in measurements. ***

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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