ijspace: Mass Length Topology

Previously we introduced a constant j_ML 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 j_ML, is time-independent. (Topology represents the path connectivity in measurement space, we will discuss it in a later blog.)

We then discussed Compton scattering and introduced Compton wavelength λ_Comptonwhich is independent of the energy of the photon being scattered, and hence in our context j_MLonce expressed in terms of λ_Compton, is independent of the measurement apparatus. The constant j_ML should have the same value for elementary particles as well as macroscopic structures such as black holes. Therefore if we calculated:

MLT Eq1a

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 particles¹:

Particle	Rest mass m_o	Compton wavelength λ_Compton	Calculated value of j_ML
Electron	9.109 382 91 × 10^-31	2.426 310 2389 × 10^-12	2.210 218 902 × 10^-42
Muon	1.883 531 475 × 10^-28	11.734 441 03 × 10^-15	2.210 218 902 × 10^-42
Tau	3.167 47 × 10^-27	0.697 787 × 10^-15	2.210 219 389 × 10^-42

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 time-independent constant j_ML, does not change.

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:

MLT eq2

We move into the topological space by making the metric g_Sch time independent, i.e. we equate the coefficient of dt² to zero³. This gives us, Schwarszchild radius as,

R_Sch is the radius defining the event-horizon of a black hole, and M_BH is the mass of the black hole. We can define the corresponding wavelength as,

MLT_Eq3

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 j-space, we can write,

Substituting for λ_Sch in above expression, we obtain the expression for Planck mass,

MLT_Eq4c .

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 on 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 Kruskal-Szekeres 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 region-I. The Planck mass M_P lies in the interior region-II. So how do we reconcile two very different values for the measurements of mass of same topological object?

Kruskal-Szekeres representation
in j-space

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 region-I, 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 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 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 region-II,

MLT Eq4c .

In order to move to the topology in the exterior region-I, we can replace M_BH with the Zero-Entropy mass m_ZE, measured by the maximum efficiency observer Obs _c. Recall that the zero-entropy observer means the condition ds² = 0_j, where ds is the space-time interval given as ds² = g_μνds^μds^ν, and therefore mass m_ZEcorresponds to the condition for the space-time interval equal to zero (0_j). The relationship becomes,

MLT Eq4d

We note that on the right hand side we have universal constants h and c, along with zero-entropy 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 region-I. Therefore in exterior region-I, G is expressed in terms constants h, c, and m_ZE, each of which is a universal constant for the exterior region-I, and hence G itself is a universal constant for the exterior region-I (only), denoted by G_I. Note that G_I is not G_Newton, the known universal constant of gravitation. (Note: Knot-Theory describes exterior region-1.)

Similarly to derive the relationship for G in the region-II, we replace m_ZE with m_p , and use the relationship G_II = hc÷(2πm_p²), where m_p is Planck mass. At this point we can not be sure that m_ZEand m_p are equal, unless they can be precisely measured. (More importantly the value of G in the region-IV G_IV , can not assumed to be equal to G_II .)

What if there was another observer, Obs_i from higher information space⁴, whose measurement Hamiltonian is 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 zero-entropy 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 shallow-well (just another inconvenient pot-hole) or more formidable infinite-potential 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.

The black hole exists in Aku's reference frame, because the topology of the information space can not be measured with precision. Thus a very high measured value of mass corresponding to an infinitesimal information content δ_i, is specific to the exterior region-I⁵. If that is the case, then high value of mass measured in region-I for a specific information content, is an intrinsic property specific to the entropy of the region-I only.

We already know that the entropy of region-I is an invariant, in fact the entropy is the only invariant in the relativistic thermodynamic description of region-I. Then what is the relationship between the entropy of the measurement and the mass? Do we have to introduce information-mass besides inertial and gravitational masses? We will discuss this rather fascinating topic some other time. Meanwhile perhaps we can speculate on the actual mass of the universe itself and how to measure it with absolute precision?

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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 0_j in discrete measurement space. More discussion can be found here.

We will also like to point out that by making metric g_Sch time independent, the coefficient of the dr² term, (~|c²/0_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 dθ² and dφ², i.e. rotational components only, with a massive singularity accompanying dr² 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.

4. We remember that the electron-photon interaction corresponds to q = 3 space. The higher information spaces are either q = 1 or q = 2.

5. The infinitesimal information content δ_i, is the only true invariant in all four regions of space-time in Kruskal-Szekeres coordinates.

* Zero-Entropy mass: Observer Obs_i - Topological Space, Shallow well, unknot. Inertial mass: Observer(s) Obs_M(s) - Manifold(s), Box, knot(s) (Trefoil for most efficient Observer in j-space ≡ Obs_c). And then there is mass corresponding to measurements made for a δ-function deep in the region-IV of Kruskal-Szekeres coordinates. What should we call it?

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