January11, 2016
Hiroshi Michiwaki
Transition mechanisms of mass derived from the special theory of relativity andreducible set theoretical division by zero
1. Introduction
In the section titled “III. Discussion about the special theory of relativity and reducible
set theoretical division by zero” in the note from October 10, 2014 titled “Research on
the derivation of a parameter that connects physical equations and reducible set
theoretical division by zero,” we showed that when a finite mass (hereafter referred toas a massive particle) travels at the speed of light, the apparent relativistic mass m
becomes 0, while on the other hand, the rest mass m0 remains as a system. This wasderived using reducible set theoretical division by zero described in the note fromMarch 18, 2014.
In this note, we describe this interpretation in further detail. It is thought that A.Einstein
believed that objects that travel at speeds greater than the speed of light c do not exist,
except for light, based on his construction of the special theory of relativity and hisinterpretation of the Lorentz factor (refer to Equation (1)) in the Lorentz transformation.
This is because he believed that at the limit v → c, γ → ∞, as shown in Equation (1):

It is believed that Einstein had no choice but to arrive at the interpretation that massive
particles can never travel at the speed of light in order to avoid this illogical result. Thisis because the relativistic mass m is given by the following equation:

According to reducible set theoretical division by zero, when v = c in Equation (2), weobtain

Here, the symbol “...” on the rightmost side of Equation (3) means that everything after“...” is a remainder term.
2. The implications of Equation (3)Equation (3) states that when an object travels at the speed of light c, the apparent
relativistic mass m disappears and becomes 0, and that the rest mass m0 appears as asystem. In other words, this expression states that an object that is travelling at the speedof light c is observed as a rest mass m0.
We apply this statement to Equation (3) using a photon as an example. Because the rest
mass cm0 of a photon iswe obtain.\According to Equation (5), light (a photon) has no mass, regardless of whether it is atrest, traveling at the speed of light c, or traveling at a speed other than the speed of lightc. In other words, its mass is invariably 0.
Next, consider a neutrino. Neutrinos are believed to have a finitely small rest mass vm0(below several eV/c2). Based on this, the mass can be expressed as :Furthermore, for a speed of travel v that satisfies 0 < v < c,holds. Furthermore, at the limit v→c,

In the local point v = c, we obtainIn other words, this implies that for a neutrino traveling at the speed of light c, therelativistic effect disappears, and as a result,is observed.
It is thought that the above result is the mathematical mechanism in the special theory ofrelativity that follows from reducible set theoretical division by zero for mass changes
for massive particles such as neutrinos.
This relationship is shown in the graph in Fig. 1. The light blue line in Fig. 1
corresponds to a graph of the Lorentz factor. As can be seen from this graph, therelativistic effect appears until v → c, and when c = v, a strong discontinuity appears
similar to division by zero, and vm(c) = vm(0) = vm0.
In other words, the most natural interpretation in reducible set theoretical division by
zero is to conclude that for a massive particle traveling at the speed of light c, therelativistic effect disappears, and the original mass m0 remains or appears.

Fig. 1 The relationship between the speed of travel and the mass of a neutrino
If the speed is reduced from the speed of light c, then the apparent mass vm(v2) will
increase so that it satisfies vm(v2) > vm0 > 0. However, if the speed is reduced evenfurther, then the apparent mass vm(v1) will decrease so that it satisfies vm(v2) > vm(v1) >vm0 > 0 (v1 < v2) following the graph of the Lorentz factor.
It is important to capture the manner in which the speed decreases, and the mechanismof mass transition. It is possible that the mass transition is occurring with adiscontinuous value. As a result, the infinite mass seen in Equation (8) does not appear
in reality, and when the speed of the neutrino decreases from the speed of light c, it does
not take on a value that is a limit value such as v→c, and decreases through appropriate
discrete values.

3. Note about the creation of Fig. 1
Optical observations and neutrino observations of the light that arrived at Earth on
February 23, 1987 from the supernova explosion that was emitted from the Large
Magellanic Cloud, which is 160,000 light-years away, as well as the observation resultsof the supernova explosion from M101, which is an astronomical phenomenon which
was discovered on August 23, 2011 and which reached its peak on September 13,
showed that neutrinos can escape supernova explosion regions more easily than light
can, that neutrinos were observed preceding light in optical observations by 3 h, and thatneutrinos and light that traveled to the Earth from 27,000,000 light-years away reachedthe Earth with almost no time difference. This observation shows that if it is assumed
that there is a difference of only a few seconds in the time it takes for an object travelingat the speed of light and an object travelling at the speed of a neutrino to reach anobservation point on the Earth from a supernova explosion, then an object travelling at
the speed of a neutrino is able to travel at approximately 99.999999999999% of thespeed of light. In this case, the value of the Lorentz factor in the Lorentz transformation
becomes approximately 7.07 × 106, and the mass of a neutrino, which is less thanseveral eV, will be observed as several MeV.
The following assumptions are made in the creation of Fig. 1.
i. A neutrino has a finite rest mass vm(0) =vm0>0. If the rest mass is vm(0) =vm0=0, then aneutrino will not exhibit any mass at any speed, similar to a photon. This would
contradict the observed reality.
ii. The speed of light ratio of the neutrino travel speed was calculated from Table 2
based on assumptions about the difference in delay between the optical observations andthe neutrino observations when the two supernova explosions (the Large Magellanic
Cloud and M101-NGC5457) were observed, and based on the calculated value of thedifference in delay in the neutrinos that traveled from the farther supernova explosion inparticular. The difference in delay between the optical observations and the neutrino
observations is assumed to be no more than several seconds, and the fact that theneutrinos, which are believed to be able to escape from the supernova explosion region
more easily than light, arrived approximately 3 h earlier is ignored. Therefore, based on
Table 2, a value of 99.999999999% is used for the speed of light ratio, whichcorresponds to a difference in delay of 8.46 s.
iii. The marker for vm(0.99999999999999c) on the graph uses the value of the Lorentz
factor that corresponds to the speed of light ratio selected from Table 2 and applied toTable 3.
iv. The marker for vm(0.9999999999c) on the graph plots the value of the ratio of the
published value for the upper limit of the mass of a muon neutrino to the publishedvalue for the upper limit of the mass of an electron neutrino corresponding to the valueof the Lorentz factor in Table 3.

v. The marker for v = c on the graph uses the result vm(c) = vm(0) = vm0 > 0 in Equation(9) and Equation (10).
Table 1: Relationship between neutrino speed v and number of seconds delay t in theLarge Magellanic Cloud supernova explosion*
Speed of light ratio of
neutrino travel speed

Required travel
time [s]Requirednumber ofyears

Number ofyears ofdelay

Number of days ofdelay

Number of seconds of delay*Distance to the Large Magellanic Cloud: 160,000 light-years, 1.51372 × 1021 m

Table 2: Relationship between neutrino speed v and number of seconds delay t in theM101 (NGC 5457) supernova explosion*
Speed of light ratio ofneutrino travel speed

Required travel
time [s]Requirednumber ofyears

Number ofyears ofdelay

Number of days ofdelay

Number of seconds of delay*Distance to M101 (NGC 5457): 27,000,000 light-years, 2.5544 × 1023 m
Table 3: Relativistic mass magnification, calculated values of Lorentz factor
Speed of light ratio of neutrino
travel speed

Speed of light squared(v^2/c^2)

1-(v^2/c^2)√(1-(v^2/c^2))1/√(1-(v^2/c^2))