Malcolm J. Macleod

Following the 26th General Conference on Weights and Measures are fixed the numerical values of the 4 physical constants ($h, c, e, k_B$). This is premised on the independence of these constants. This article discusses a model of a mathematical electron from which can be defined the Planck units as geometrical objects (mass M=1, time T=2$\pi$ ...). In this model these objects are interrelated via this electron geometry such that once we have assigned values to 2 Planck units then we have fixed the values for all Planck units. As all constants can then be defined using geometrical forms (in terms of 2 fixed mathematical constants, 2 unit-specific scalars and a defined relationship between the units $kg, m, s, A$), the least precise CODATA 2014 constants ($G, h, e, m_e, k_B$...) can then be solved via the most precise ($c, \mu_0, \alpha, R_\infty$), with numerical precision limited by the precision of the fine structure constant $\alpha$. In terms of this model we now for example have 2 separate values for elementary charge, calculated from ($c, \alpha, R_\infty$) and the 2017 revision.

The simulation hypothesis proposes that all of reality is an artificial simulation. In this article I describe a simulation model that derives Planck level units as geometrical forms from a virtual (dimensionless) electron formula $f_e$ that is constructed from 2 unit-less mathematical constants; the fine structure constant $\alpha$ and $\Omega$ = 2.00713494... ($f_e = 4\pi^2r^3, r = 2^6 3 \pi^2 \alpha \Omega^5$). The mass, space, time, charge units are embedded in $f_e$ according to these ratio; ${M^9T^{11}/L^{15}} = (AL)^3/T$ (units = 1), giving mass M=1, time T=$2\pi$, length L=$2\pi^2\Omega^2$, ampere A = $(4\pi \Omega)^3/\alpha$. We can thus for example create as much mass M as we wish but with the proviso that we create an equivalent space L and time T to balance the above. The 5 SI units $kg, m, s, A, K$ are derived from a single unit u = sqrt(velocity/mass) that also defines the relationships between the SI units; kg = $u^{15}$, m = $u^{-13}$, s = $u^{-30}$, A = $u^{3}$, $k_B = u^{29}$. To convert MLTA from the above $\alpha, \Omega$ geometries to their respective SI Planck unit numerical values (and thus solve the dimensioned physical constants $G, h, e, c, m_e, k_B$) requires an additional 2 unit-dependent scalars. Results are consistent with CODATA 2014. The rationale for the virtual electron was derived using the sqrt of momentum P and a black-hole electron model as a function of magnetic-monopoles AL (ampere-meters) and time T.

Outlined here is a simulation hypothesis approach that uses an expanding (the simulation clock-rate measured in units of Planck time) 4-axis hyper-sphere and mathematical particles that oscillate between an electric wave-state and a mass (unit of Planck mass per unit of Planck time) point-state. Particles are assigned a spin axis which determines the direction in which they are pulled by this (hyper-sphere pilot wave) expansion, thus all particles travel at, and only at, the velocity of expansion (the origin of $c$), however only the particle point-state has definable co-ordinates within the hyper-sphere. Photons are the mechanism of information exchange, as they lack a mass state they can only travel laterally (in hypersphere co-ordinate terms) between particles and so this hypersphere expansion cannot be directly observed, relativity then becomes the mathematics of perspective translating between the absolute (hypersphere) and the relative motion (3D space) co-ordinate systems. A discrete `pixel' lattice geometry is assigned as the gravitational space. Units of $\hbar c$ `physically' link particles into orbital pairs. As these are direct particle to particle links, a gravitational force between macro objects is not required, the gravitational orbit as the sum of these individual orbiting pairs. A 14.6 billion year old hyper-sphere (the sum of Planck black-hole units) has similar parameters to the cosmic microwave background. The Casimir force is a measure of the background radiation density.

The Simulation Hypothesis proposes that all of reality is in fact an artificial simulation, analogous to a computer simulation. Outlined here is a method for programming relativistic mass, space and time at the Planck level as applicable for use in Planck Universe-as-a-Simulation Hypothesis. For the virtual universe the model uses a 4-axis hyper-sphere that expands in incremental steps (the simulation clock-rate). Virtual particles that oscillate between an electric wave-state and a mass point-state are mapped within this hyper-sphere, the oscillation driven by this expansion. Particles are assigned an N-S axis which determines the direction in which they are pulled along by the expansion, thus an independent particle motion may be dispensed with. Only in the mass point-state do particles have fixed hyper-sphere co-ordinates. The rate of expansion translates to the speed of light and so in terms of the hyper-sphere co-ordinates all particles (and objects) travel at the speed of light, time (as the clock-rate) and velocity (as the rate of expansion) are therefore constant, however photons, as the means of information exchange, are restricted to lateral movement across the hyper-sphere thus giving the appearance of a 3-D space. Lorentz formulas are used to translate between this 3-D space and the hyper-sphere co-ordinates, relativity resembling the mathematics of perspective.

The Simulation Hypothesis proposes that all of reality, including the earth and the universe, is in fact an artificial simulation, analogous to a computer simulation, and as such our reality is an illusion. In this essay I describe a method for programming mass, length, time and charge (MLTA) as geometrical objects derived from the formula for a virtual electron; $f_e = 4\pi^2r^3$ ($r = 2^6 3 \pi^2 \alpha \Omega^5$) where the fine structure constant $\alpha$ = 137.03599... and $\Omega$ = 2.00713494... are mathematical constants and the MLTA geometries are; M = (1), T = ($2\pi$), L = ($2\pi^2\Omega^2$), A = ($4\pi \Omega)^3/\alpha$. As objects they are independent of any set of units and also of any numbering system, terrestrial or alien. As the geometries are interrelated according to $f_e$, we can replace designations such as ($kg, m, s, A$) with a rule set; mass = $u^{15}$, length = $u^{-13}$, time = $u^{-30}$, ampere = $u^{3}$. The formula $f_e$ is unit-less ($u^0$) and combines these geometries in the following ratio M$^9$T$^{11}$/L$^{15}$ and (AL)$^3$/T, as such these ratio are unit-less. To translate MLTA to their respective SI Planck units requires an additional 2 unit-dependent scalars. We may thereby derive the CODATA 2014 physical constants via the 2 (fixed) mathematical constants ($\alpha, \Omega$), 2 dimensioned scalars and the rule set $u$. As all constants can be defined geometrically, the least precise constants ($G, h, e, m_e, k_B$...) can also be solved via the most precise ($c, \mu_0, R_\infty, \alpha$), numerical precision then limited by the precision of the fine structure constant $\alpha$.

Defined are gravitational formulas in terms of Planck units and units of $\hbar c$. Mass is not assigned as a constant property but is instead treated as a discrete event defined by units of Planck mass with gravity as an interaction between these units, the gravitational orbit as the sum of these mass-mass interactions and the gravitational coupling constant as a measure of the frequency of these interactions and not the magnitude of the gravitational force itself. Each particle that is in the mass-state (defined by a unit of Planck mass) per unit of Planck time is directly linked to every other particle also in the mass-state by a discrete unit of $m_P v^2 r = \hbar c$, the velocity of a gravitational orbit is summed from these individual $v^2$. As this approach presumes a digital time, it is suitable for use in programming Simulation Hypothesis models. As this link is responsible for the particle-particle interaction it is analogous to the graviton. Orbital angular momentum of the planetary orbits derives from the sum of the planet-sun particle-particle orbital angular momentum irrespective of the angular momentum of the sun itself and the rotational angular momentum of a planet includes particle-particle rotational angular momentum.