Programming a Virtual (Mathematical) Universe

Our external physical reality is a mathematical structure. That is, the physical universe is mathematics in a well-defined sense, and in those [worlds] complex enough to contain self-aware substructures SAS [they] will subjectively perceive themselves as existing in a physically 'real' world (- Max Tegmark's MUH).

Summary: cont from previous page. uses black-hole electron to separate the Planck units into a dimensionless geometrical component (alpha, Omega), a dimensioned unit (u) and 2 scalars (x, y).

A rotating charged Dirac Kerr-Newman black-hole electron

A charged rotating black hole is a black hole that possesses angular momentum and charge. In particular, it rotates about one of its axes of symmetry. In physics, there is a speculative notion that if there were a black hole with the same mass and charge as an electron, it would share many of the properties of the electron including the magnetic moment and Compton wavelength. This idea is substantiated within a series of papers published by Albert Einstein between 1927 and 1949. In them, he showed that if elementary particles were treated as singularities in spacetime, it was unnecessary to postulate geodesic motion as part of general relativity. The Dirac–Kerr–Newman black-hole electron was introduced by Burinskii using geometrical arguments. The Dirac wave function plays the role of an order parameter that signals a broken symmetry and the electron acquires an extended space-time structure. Although speculative, this idea was corroborated by a detailed analysis and calculation.

A black-hole 2-state electron model

In this model the electron is envisaged as an oscillating-over-time 2-state event comprising an electric (magnetic monopole) state and a (Planck) mass micro black-hole state.

1. The electric state is constructed from magnetic monopoles. The units for a monopole are the ampere-meter (AL = ampere length), the electric state is a function of (AL)^3 and time T; the duration or period (frequency) of the electric state is determined by the electron function fe =  (AL)^3/T. In this model the dimensions of charge, mass, space and time are not independent but overlap and in a particular ratio will cancel. This ratio is found in the electron function fe, as such fe (the electric state) is a dimensionless mathematical constant (its numerical value is independent of the system of units used).

2. The mass state comprises a Planck size micro black-hole; Planck mass = 1, Planck time = 1.

Units for M^9T^11/L^15 = (AL)^3/T = 1
fe = (AL)^3/T = 0.12692 x 10^23; units = 1

SI units as array structures in terms of u

The are 5 principal SI units; (kg, s, m, A=ampere (charge), K=kelvin or celsius) ... see sqrt of Planck momentum. These are defined here in terms of a dimensioned array u as exponents of the form un. Thus only 1 dimensioned unit is required.


SI dimensioned physical constants as geometrical forms

There are 6 fundamental dimensioned physical constants; (G, h, e, c, me, kB). These are defined in terms of 2 mathematical (dimensionless) constants;  the fine structure constant alpha and a proposed Omega and assigned a dimensioned unit of the form un. 2 scalars are used to translate from the base geometries of alpha and Omega to the SI numerical values. The geometry appears to encode the function of the constant. Thus the physical constants (G, h, e, c, me, kB) can be defined using 2 mathematical constants, 2 scalars and 1 dimensioned unit, see calculator below.


Underlying sub structure beta 

A primitive geometry denoted beta is constructed from 1 unit of Omega, 2 scalars and the unit u. From this geometry we can construct the physical constants as exponential mulitples


The formulas referenced here are listed in the following article (see also sqrt of Planck momentum);

The pdf has the maple code for the article formulas.

Alpha* as input (0 = default)



scalar p      
scalar v      
Omega       

Solutions (α Ω v, p) :

Planck constant*

h = 6.626 070 040(81)e-34


elementary charge*

e = 1.602 176 6208(98)e-19


electron mass*

me = 9.109 383 56(11)e-31


electron wavelength*

v = 2.4263102367(11)e-12


Boltzmann constant*

kB = 1.380 648 52(79)e-23


Proton mass*

mp = 1.672621777(74)e-27


Neutron mass*

mn = 1.674927351(74)e-27


Atomic mass unit*

mu = 1.660538921(73)e-27


von klitzing*

RK = 25 812.807 4555(59)


magnetic flux*

fx0 = 2.067 833 831(13)


Gravitation constant*

G = 6.674 08(31)e-11


electron gyro*

Ge = 1.760 859 644(11)e11


Planck mass*

mP = 2.176 470(51)e-8


Planck length*

lp = 1.616 229(38)e-35


Planck time*



Bohr magneton*

9.274 009 994(57) e-24


Bohr radius*

a0 = 0.52917721067(12) e-10


Gravitational coupling constant

aG = 1.7518e-45


Q (sqrt of Planck momentum)

The calculator (left) calculates Omega and the scalar units p and v. It then uses these 4 values; alpha, Omega, p, v to solve the constants as per the formulas below. The solutions are then compared with the CODATA 2014 equivalents for comparison.

The value for alpha changes with each codata update and so I have selected it as user input; the default value 0 = 137.035999139 (codata 2014 mean)

From MLTVPA, we can derive G, h, c, e, me, kB...:
Planck units from α, Ω, p, v (maple format);
P:=Omega*p:  
V:=2*pi*Omega^2*v:  

MTLA using p,v:
T:=2*pi*p^(9/2)/v^6: 
M:=(2*pi*P^2/V): 
L:=(T*V/2):  
A:=(8*V^3/(a*P^3)): 

Physical constants from MLTVPA
sigma:=(pi^2/(3*a^2*A*L)): (magnetic monopole)
fe:=(T*sigma^3): (electron function)
me=M*fe; (electron mass)
Tp=(A*V/pi); (Planck temperature)
mu0=pi*M*V^2/(a*L*A^2); (permeability of vacuum)
e=(A*T); (elementary charge)
h=(2*pi*L*M*V); (Planck constant)
kB=pi*M*V/A; (Boltzmann constant)
G=(V**2*L/M); (Gravitation constant)

Planck units as alpha, Omega, k (mass), t (time);
M:=1*k:
T:=2*pi*t:
P:=Omega*k^(4/5)/t^(2/15):
V:=2*pi*P^2/M:
L:=(T*V/2):
A:=(8*V^3/(a*P^3)):

Planck units as alpha, Omega, z^3 (ampere), l (length);
A:=(64*pi^3*Omega^3/a)*z^3:
L:=(2*pi^2*Omega^2)*l:
T:=(2*pi)*z^9*l^3:
V:=(2*L/T):
M:=(2^3*pi*V)/(a^(2/3)*A^(2/3)):

Notes:
- listed reference values from CODATA 2014
- proton mass from electron-proton ratio CODATA mean
- neutron mass from electron-neutron ratio CODATA mean

Comment:

As referenced in the article, physics still debates the number of dimensioned units that are required, the present consensus being 3; MLT of the 5 SI units (kg, m, s, Ampere, kelvin), nevertheless the 6 principal physical constants  G, h, c, e, me, kB  cannot be derived in terms of each other (although they are defined using the 5 SI units) and are supposed to be independent of each other (physics has no model which may link these physical constants together hence the designation as fundamental constants).

This model suggests that the geometry of the constant confers the attributes of its units. There is an associated unit u which is the dimensional component of the constant but as the information of unit u is embedded within the formula for the constant itself, u has a descriptive rather than a physical role. Consequently I argue that it is possible to construct our physical mass space time units in a virtual (mathematical) environment.