virtual black-hole electron and the simulation hypothesis Programming Planck units from a virtual (black-hole) electron; a Simulation Hypothesis Malcolm J. Macleod [email protected] The simulation hypothesis proposes that all of reality is an artificial simulation. In this article I describe a programmable simulation method that constructs the Planck units as geometrical forms derived from a virtual 2 3 6 2 5 electron fe ( fe = 4π r ; r = 2 3π αΩ , units = 1), itself a function of 2 unit-less mathematical constants; the fine structure constant α and Ω = 2.0071349496... The Planck units are embedded in fe according to these ratios; M9T 11=L15,(AL)3=T ... units = 1; giving geometries mass M=1, time T=2π, velocity V = 2πΩ2, length L=2π2Ω2 ... 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 in accordance with these ratio. The 5 SI units kg; m; s; A; K are replaced by a single unit u that defines the relationships between the SI units; kg = u15, m= u−13, s = u−30,A = u3 ... The units for u are sqrt(velocity/mass). To convert from base (Planck) geometries to their respective SI numerical values requires 2 dimensioned scalars with which we can then solve the SI physical constants G; h; e; c; me; kB. Results are consistent with CODATA 2014 (see table, numerical scalars k; v from mP and c). The rationale for the virtual electron was derived via the sqrt of Planck momentum and a black-hole electron model as a function of magnetic-monopoles AL and time T. In summary we can reproduce both the physical constants G; h; e; c; me; kB and the SI units kg; m; s; A; K via a this virtual electron using 2 mathematical constants (α, Ω), 2 dimensioned scalars (whose numerical values depend on the system of units used) and 1 dimensioned unit u. Table 1 Calculated∗ (α, Ω; k; v) CODATA 2014 Speed of light V = 299792458 u17 c = 299792458 ∗ 7 56 7 Permeability µ0 = 4π/10 u µ0 = 4π/10 ∗ 13 Rydberg constant R1 = 10973731.568 508 u R1 = 10973731:568 508(65) [15] Planck constant h∗ = 6:626 069 134 e-34 u19 h = 6:626 070 040(81) e-34 [16] Elementary charge e∗ = 1:602 176 511 30 e-19 u−27 e = 1:602 176 6208(98) e-19 [19] ∗ 15 Electron mass me = 9:109 382 312 56 e-31 u me = 9:109 383 56(11) e-31 [17] ∗ 29 Boltzmann’s constant kB = 1:379 510 147 52 e-23 u kB = 1:380 648 52(79) e-23 [22] Gravitation constant G∗ = 6:672 497 192 29 e-11 u6 G = 6:674 08(31) e-11 [21] keywords: computer universe, virtual universe, mathematical universe, simulated universe, sqrt Planck momentum, Planck unit, magnetic-monopole, fine structure constant alpha, Omega, black-hole electron; 1 Background tp, Planck length lp, Planck charge AQ, Planck temperature T ) are a set of units of measurement also known as natural The general universe simulation hypothesis proposes that all P units because the origin of their definition comes only from of reality, including the earth and the universe, is in fact an properties of nature and not from any human construct. artificial simulation, analogous to a computer simulation, and as such our reality is an illusion [2]. 2 The virtual universe Mathematical platonism is a metaphysical view that there are abstract mathematical objects whose existence is indepen- Mathematical universe hypotheses presume that our physical dent of us [1]. Mathematical realism holds that mathematical universe has an underlying mathematical origin. The princi- entities exist independently of the human mind. Thus humans pal difficulty of such hypotheses lies in the problem of con- do not invent mathematics, but rather discover it. Triangles, structing physical units such as mass, space and time from for example, are real entities, not the creations of the human their respective mathematical forms. mind [3]. This article describes a mathematical universe model that 2 3 6 2 5 Max Tegmark’s Mathematical Universe Hypothesis: Our is based on a virtual electron ( fe = 4π r ; r = 2 3π αΩ ) external physical reality is a mathematical structure. That is, from which the Planck units can be derived as geometrical the physical universe is mathematics in a well-defined sense, forms. The fine structure constant α, π and a recurring con- and ”in those [worlds] complex enough to contain self-aware stant Ω are dimensionless mathematical constants, thus the substructures [they] will subjectively perceive themselves as electron formula fe is also a mathematical constant, and as existing in a physically ’real’ world” [10]. such has a numerical solution that is independent of the sys- 23 Planck units (defined here Planck mass mP, Planck time tem of units used ( fe = 0:2389545x10 ). 1 2 The virtual universe virtual black-hole electron and the simulation hypothesis From this electron formula we can derive the Planck units 2πQ2 q2 Fp = ; units = (5) as geometrical forms; mass M=1, time T=2π, velocity V = tp s 2πΩ2, length L=2π2Ω2 ... (4.1). 3.2. The charge constants in terms of Q3; c; α, l ; The 5 SI units kg; m; s; A; k are replaced by a single unit p u (5.0) which defines the relationships between the SI units. 8c3 m3 q3 p A = ; unit A = = (6) The units for u are velocity=mass. The functionality of each Q αQ3 q3 s3 kg3 (Planck) unit is embedded into its α, Ω geometry. 3 2 3 To solve the physical constants in SI terms also requires 8c 2lp 16lpc q s e = A t = : = ; units = A:s = (7) 2 scalars to convert from these base geometries to their re- Q p αQ3 c αQ3 kg3 spective SI values (4.2), i.e.: using the following values for A c 8c3 c 8c4 q5 α, Ω; k; v gives the results in the table (p1) for G; h; c; e; me; kB; Q T p = = : = ; units = (8) −7 15 3 3 4 k = mP = :2176728175:::x10 u (kg) π αQ π παQ kg 2 17 2 5 3 v = (2πΩ )=c = 11843707:9::: u (m/s) Ep π αQ kg k ; units α = 137:035999139 CODATA mean (4.4) B = = 3 = (9) T p 4c q Ω = 2:0071349496...; (4.5) Thus we may construct our physical mass, space and time 3.3. As with c, the permeability of vacuum µ0 has been as- using 2 dimensionless constants (α, Ω), 2 dimensioned scalars signed an exact numerical value so it is our next target. The and a single unit u using mathematical forms derived from a ampere is that constant current which, if maintained in two virtual electron. straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, 3 Sqrt of Planck momentum would produce between these conductors a force equal to ex- actly 2:10−7 newton per meter of length. In this section I introduce the sqrt of momentum as a distinct 2 3 8 Planck unit and use this to link the mass and the charge con- F Fp 1 2πQ αQ παQ 2 electric = : = :( )2 = = (10) stants. From the formulas for the charge constants I derive 2 2 3 5 7 AQ α AQ αtp 8c 64lpc 10 a formula for a magnetic monopole (ampere-meter AL) and from this a formula for an electron fe. I then argue that fe π2αQ8 4π kg:m kg6 µ = = ; units = = (11) is dimensionless as the monopole and time units are not in- 0 5 7 2 2 4 32lpc 10 s A q s dependent but rather overlap, collapsing within the electron 3 3.4. Rewritting Planck length lp in terms of Q; c; α, µ0; according to this ratio; fe = (AL) =T , units = 1. Being di- mensionless and so independent of any system of units, this π2αQ8 q2 s electron formula is a mathematical constant. l = ; unit = = m (12) p µ c5 kg Note: for convenience I use the commonly recognized 32 0 ∼ value for alpha as α 137:036 3.5. A magnetic monopole in terms of Q; c; α, lp; The ampere-meter is the SI unit for pole strength (the Defining Q as the sqrt of Planck momentum where Planck product of charge and velocity) in a magnet (Am = ec). A 2 momentum = mPc = 2πQ = 6.52485... kg:m=s, and a unit q magnetic monopole σe is a hypothetical particle that is a mag- 2 whereby q = kg:m=s giving; net with only 1 pole [12]. I propose a magnetic monopole σe −6 from α, e; c (σe = 0:13708563x10 ); Q = 1:019 113 411:::; unit = q (1) 3α2ec q5 s σ = ; units = (13) Planck momentum; 2πQ2; units = q2, e 2π2 kg4 2 Planck length; lp; units = m = q s=kg, c; units = m=s = q2=kg; I then use this monopole to construct an electron frequency 23 3.1. In Planck terms the mass constants are typically defined function fe ( fe = 0:2389545x10 ); in terms of Planck mass, here I use Planck momentum; 3 8 3 3 2 10 3 5 7 15 2 σ 2 3 α lpc 3 α Q q s f = e = = ; units = (14) e 6 9 2 2 12 2πQ2 tp π Q 4π µ kg m = ; unit = kg (2) 0 P c 3.6. The most precisely measured of the natural constants is 2 4 the Rydberg constant R1 (see table) and so it is important to 2 2 kg:m q Ep = mPc = 2πQ c; units = = (3) this model.