|Home||Mass, Length, & Topology|
|19th May 2019|
"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.
"Show that the theory defined on this book is fully consistent; if it is not, correct it.
Find observable and testable consequences.
Using these, test theory.
If the theory is confirmed, we will feel good, and feeling good is good enough..."
- Carlo Rovelli and Francesca Vidotto, Exercise 13.1 in Covariant Loop Quantum Gravity: An elementary introduction to quantum gravity and spinfoam theory.
Previously we introduced a constant jML with dimensions of Mass-Length, 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 j-space, in which the topology is defined as the entropy-free space. The entropy-free space implies that time will not be a factor. Thus the constant jML, is time-independent. (Topology represents the path connectivity in measurement space, we will discuss it in a later blog.)
where mo is the rest mass for elementary particles, the value of jML should come out to be the same for every elementary particle. The following table shows the calculated value of jML for some of the elementary particles1:
No surprises here, as λCompton'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 time-independent constant jML, 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 λCompton.)
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 gSch time independent, i.e. we equate the coefficient of dt2 to zero3. This gives us, Schwarszchild radius as,
RSch is the radius defining the event-horizon of a black hole, and MBH is the mass of the black hole. We can define the corresponding wavelength as,
Subsequently using the criterion that the constant jML, representing the inherent topology of a given measurement space, does not change for all structures in j-space, we can write,
Substituting for λSch in above expression, we obtain the expression for Planck mass,
The estimated mass of a black-hole is of the order of solar mass, approximately 2 × 1030 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.
If we recall, RSch 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 region-I, is no longer applicable. Aku or ObsM 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 region-I, per this observer it will take Aku forever to enter the black hole. Thus it is impossible for an observer in the region-I 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 electron-photon interaction (q = 3 space) based measurement apparatus available to Aku. Even if Aku was inside RSch, 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.
In order to move to the topology in the exterior region-I, we can replace MBH with the Zero-Entropy mass mZE , measured by the maximum efficiency observer Obsc. Recall that the zero-entropy observer means the condition ds2 = 0j, where ds is the space-time interval given as ds2 = gμν dsμdsν, and therefore mass mZE corresponds to the condition for the space-time interval equal to zero (0j). The relationship becomes,
1. Based on the values for Compton wavelengths from National Institute of Standards and Technology (NIST). The units are in MKS.
2. The condition of h/c = 1 rather than h = 1 and c = 1 raises an interesting scenario. In the measurement space of a macroscopic observer, entropy and hence corresponding time-axis are bound to exist. However it is also plausible that values of the fundamental constants would also change. Since these constants relate to Planck space, a macroscopic observer need to perform experiments at the edge of the universe, to detect variations in their values. If such measurements were possible, the constants h and c, are likely to show similar variations, variations which are not independent of each other.
3. In discrete measurement space, we are not interested in the removal of the geometrical singularity at the origin. It is understood that a certain region smaller than the Least Energy Surface, will remain inaccessible to Aku and hence could not be measured by Aku. We note that the least energy surface provides with the definition of origin 0j in discrete measurement space. More discussion can be found here. 4. We remember that electron-photon interaction corresponds to q = 3 space. The higher information spaces are either q = 1 or q = 2. 5. The infinitesimal information content δi, is the only true invariant in all four regions of space-time in Kruskal-Szekeres coordinates.
A Timeless Constant
Space Time and Entropy
Curve of Least Disorder
Möbius & Lorentz Transformation - II
Möbius & Lorentz Transformation - I
Knots, DNA & Enzymes
Quantum Comp - III
Quantum Comp - II
Quantum Comp - I
Insincere Symm - II
Insincere Symm - I
Existence in 3-D
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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