Programming a Virtual (Mathematical) Universe

Update pending!!!! 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: uses black-hole electron to derive dimensionless geometrical forms for dimensioned units MLTAK

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 electron model

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

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. 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 is a dimensionless mathematical constant (its numerical value is independent of the system of units used).

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

Planck units from dimensionless mathematical (geometrical) forms 

There are 6 fundamental dimensioned physical constants; (G, h, e, c, me, kB) and these are referenced in terms of 5 SI units; M=mass kg, T=time s, L=length m, A=ampere (charge), K=kelvin or celsius (temperature)... see sqrt of Planck momentum. There are also dimensionless mathematical constants such as the fine structure constant alpha and pi whose numerical values are independent of the system of units used. Thus although the value for c = 299792458 m/s, this value depends on the defintion of the meter and the second as SI units, we could equally write c = 186280 miles/s using an imperial unit the mile, likewise any aliens we meet would have their own set of units.

I propose a universal system of units in which the Planck units are constructed from 2 mathematical constants; the fine structure constant alpha and a proposed constant Omega, and 2 scalar dimensioned (Planck) units such as mass and time or momentum and velocity etc. Via these 2 scalars we can numerically solve any system of units where local or non-terrestrial (the 2 mathematical constants have fixed numerical values), thus providing the basis for a universal system of units as fixed geometrical forms.
"we get the possibility to establish units for length, mass, time and temperature which, being independent of specific bodies or substances, retain their meaning for all times and all cultures, even non-terrestrial and non-human ones and could therefore serve as natural units of measurements..." - Max Planck.



The model is described in the following article (see also sqrt of Planck momentum);             

The pdf has the maple code for the article formulas.

The Mathematical Universe (refer article)

Using 2 fixed mathematical constants (the fine structure constant alpha and a proposed Omega), I derive dimensionless geometrical forms for mass, time, length, charge ... These mathematical forms are interlocking and so may be defined in terms of each other, here I define LVA using MLP;

To add dimensionality to these geometrical forms to construct geometrical shapes requires a dimension unit, I define the base units as mltpva; m as mass, l as length, t as time, v as velocity, a as charge, p as sqrt of momentum. We can now construct generic formulas for Planck unit equivalents;

To solve any system of units, whether local or non-terrestrial, I require only 2 scalar units; in the following examples I define MLTPVA in terms of m,t (left) and also in terms of p,v (right);
  

The physical constants as constructs of 2 mathematical constants and 2 scalar units;
  • physical constant (G, h, c, e, me, kB...) = (dimensionless α, Ω geometry) x (2 dimensioned units)

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 = alpha codata 2014 mean = 137.035999139.

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