Programming Planck Units from a Virtual (Black-Hole) Electron; a Simulation Hypothesis

virtual black-hole electron and the simulation hypothesis

Programming Planck units from a virtual (black-hole) electron; a Simulation Hypothesis

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 R∞ = 10973731.568 508 u R∞ = 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 dimensionless 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 R∞ (see table) and so it is important to 2 2 kg.m q Ep = mPc = 2πQ c, units = = (3) this model. The unit for R is 1/m. For m see eq(22); s2 kg ∞ e 4 2 3 5c5µ3 13 2lp mee µ0c 2 0 1 kg t = , unit = s (4) R∞ = = , units = = (15) p c 8h3 33πα8Q15 m q17 s3

2 3 Sqrt of Planck momentum virtual black-hole electron and the simulation hypothesis

This however now gives us 2 solutions for length m, see eq(1) most accurate constants; R, c, µ0, α. I ﬁrst convert the con- and eq(15), if they are both valid then there must be a ratio stants until they include a Q15 term which can then be re- whereby the units q, s, kg overlap and cancel; placed by eq(25). Setting unit x as;

q2 s q15 s2 q17 s3 q15 s2 kg12 m = . = ; thus = 1 (16) unit x = = 1 (26) kg kg12 kg13 kg12 q15 s2 and so we can further reduce the number of units required, for Elementary charge e = 1.602 176 51130 e-19 (table p1) example we can deﬁne s in terms of kg, q; 2 2 5 3 16lpc π Q q s e = = , units = (27) kg6 αQ3 2µ c3 kg3 s = (17) 0 q15/2 π6Q15 4π5 kg3 s q3 s e3 = = , units = = ( )3.x (28) kg6 8µ3c9 33c4α8R q6 kg3 µ = = q7/2 (18) 0 0 q4 s Planck constant h = 6.626 069 134 e-34 3.7. We ﬁnd that this ratio is embedded in that electron func- 4π4αQ10 q4 s tion fe (eq 14), and so fe is a dimensionless mathematical h = 2πQ22πl = , units = (29) p 5 constant whose function appears to be dictating the frequency 8µ0c kg of the Planck units; 4 10 10 3 21 4 3 4π αQ 3 2π µ0 kg q s 3 2 3 15 2 h = ( ) = , units = = ( ) .x σe q s 8µ c5 36c5α13R2 q18 s kg f units 0 e = ; = 12 = 1 (19) tp kg (30) Boltzmann constant kB = 1.379 510 14752 e-23 Replacing q with the more familiar m gives this ratio; π2αQ5 kg3 k = , units = (31) kg.m kg.m kg24 B 4c3 q q2 = ; q30 = ( )15 = (20) s s s4 5 3 21 3 3 π µ0 kg kg 3 kg9 s11 kB = , units = = ( ) .x (32) units = = 1 (21) 332c4α5R q18 s2 q m15 Gravitation constant G = 6.672 497 19229 e-11 Electron mass as frequency of Planck mass: 2 6 6 c lp παQ q s mP G , units = = 2 = 4 (33) me = , unit = kg (22) mP 64µ0c kg fe π3µ q6 s Electron wavelength via Planck length: G5 = 0 , units = kg4 s = ( )5.x2 (34) 22036α11R2 kg4 q2 s λ = 2πl f , units = m = (23) Planck length e p e kg π22µ9 kg81 q2 s Gravitation coupling constant: l15 = 0 , units = = ( )15.x8 (35) p 235324c35α49R8 q90 s kg m 1 α e 2 , units Planck mass G = ( ) = 2 = 1 (24) mP fe 225π13µ6 kg39 1 3.8. The Rydberg constant R = 10973731.568508(65) [15] m15 = 0 , units = kg15 = . (36) ∞ P 36c5α16R2 q30 s4 x2 has been measured to a 12 digit precision. The known precision of Planck momentum and so Q is low, however with the Electron mass me = 9.109 382 31256 e-31 solution for the Rydberg eq(15) we may re-write Q as Q15 in 10 3 27 terms of; c, µ0, R and α; 16π Rµ kg 1 m3 = 0 , units = kg3 = . (37) e 36c8α7 q30 s4 x2 25c5µ3 kg12 Q15 = 0 , units = = q15 (25) 33πα8R s2 Ampere Using the formulas for Q15 eq(25) and l eq(12) we can re- 210π33c10α3R q30 s2 q3 1 p A5 = , units = = ( )5. (38) write the least accurate dimensioned constants in terms of the Q 3 kg27 kg3 x µ0

3 3 Sqrt of Planck momentum virtual black-hole electron and the simulation hypothesis

√ 9/2 3.9. (r = q) TV p ∗ − ∗ − L = = (2π2Ω2) , unit = u16 9/2 17 5= 13 (53) There is a solution for an r2 = q, it is the radiation density 2 v5 constant from the Stefan Boltzmann constant σ; 8V3 26π3Ω3 v3 A , unit u17∗3−16∗3=3 = 3 = ( ) 3 = (54) 2π5k4 4σ αP α p σ = B , r = , units = r (39) √ 15h3c2 d c For convenience here I assign r = p, unit u16/2=8

334π5µ3α19R2 kg30 1 kg6 V2L r5 r3 = 0 , units = . = = r3 (40) G∗ = = 23π4Ω6 , u34−13−15=8∗5−17∗2=6 (55) d 53c10 q36 s5 x2 q6 s M v2 4 Base units r13 h∗ = 2πMVL = 23π4Ω4 , u15+17−13=8∗13−17∗5=19 (56) 5 4.1. The formula for the electron fe incorporates dimension- v ful quantities but itself is dimensionless. This means that its AV 27π3Ω5 v4 T ∗ = = , u3+17=17∗4−6∗8=20 (57) numerical value is a mathematical constant, independent of P π α r6 23 which set of units we may use; fe = .23895453...x10 units 7 4 3 3 ∗ 2 π Ω r 3−30=3∗8−17∗3=−27 = 1. For example we can write the formula for fe in terms of e = AT = , u (58) α v3 T= tp,A = eT,V = 2lp/T; 10 ∗ πVM α r 17+15−3=10∗8−17∗3=29 2 k = = , u (59) 3α ATV 1 B 5 3 f = ( )3 = 0.239x1023, unit = 1 (41) A 2 πΩ v e 2π2 T πV2 M α µ∗ = = r7, u17∗2+15+13−6=7∗8=56 (60) Because the numerical value is fixed and units = 1, we can 0 αLA2 211π5Ω4 look for other, non SI sets of ATV (AL) units which can also α ∗−1 = v2r7, u34+56=90 (61) be used to solve fe. In this section I describe the base units 0 29π3 MLTA in terms of a geometrical component constructed from 8π5k4 α 2 fixed dimensionless mathematical constants; the fine struc- r∗ B r, u29∗4−19∗3−17∗3=8 σ = ( 3 3 ) = 29 14 22 (62) ture constant alpha and a proposed Omega (4.5), from 2 vari- 15h c 2 15π Ω able unit-dependent scalars (from the list ktlvpa), and a rela- m 1 v5 R∗ = ( e ) = , u13 (63) 2 23 3 11 5 17 9 tional unit u that replaces the SI units (kg, m, s, A, k). 4πlpα mP 2 3 π α Ω r M = (1)k, unit = u15 (mass) (42) Scalars r, v were chosen as they can be determined directly from c, µ0 (eq 49, 60), see table p1; −30 T = (2π)t, unit = u (time) (43) c v = (64) P = (Ω)p, unit = u16 (sqrt o f momentum) (44) 2πΩ2 V = (2πΩ2)v, unit = u17 (velocity) (45) 26π6Ω4 r7 = 7 (65) L = (2π2Ω2)l, unit = u−13 (length) (46) 5 α 6 3 3 I suggested ratios where units = 1 (eq21). Setting SI Planck 2 π Ω 3 A = ( )a, unit = u (ampere) (47) unit values whereby M = mP,T = tp,L = lp,A = AQ (4.1); α 15 l15 2 2 15 15 4.2 In the previous section I used 2 base units to define the L p (2π Ω ) l 4 19 30 16 17 = = . = 2 π Ω (66) others. In this example I use P, V (p units = u , v units = u ) M9T 11 m9 t11 (1)9(2π)11 k9t11 to derive MLT A and then the physical constants. Scaling p, v P p to their SI values gives M=mP,L=lp,T=tp,V=c,P=Q l15 (.20322087−36)15 u−13∗15 = = 1 9 11 −7 9 −43 11 15∗9 −30∗11 16 k t (.217672818 ) (.171585513 ) u u P = (Ω)p, unit = u (48) (67) 3 3 3 3 6 3 3 3 2 2 3 3 3 20 14 15 2 17 A L A lp (2 π Ω ) (2π Ω ) a l 2 π Ω V = (2πΩ )v, unit = u (49) Q . = = 3 = 3 T tp (α) (2π) t α 2πP2 p2 M = = (1) , unit = u16∗2−17=15 (50) (68) V v a3l3 (.1269185923)3(.20322087−36)3 u3∗3u−13∗3 = = 1 (69) P9 t (.171585513−43) u−30 T 2 = (2πΩ)15 (51) 2πV12 If we then define MLTA in terms of PV in these ratios, we p9/2 find that P and V themselves cancel leaving only the dimen- T = (2π) , unit = u16∗9/2−17∗6=−30 (52) v6 sionless components. Consequently these ratios are unit-less,

4 4 Base units virtual black-hole electron and the simulation hypothesis

it is via this Ω15 configuration that the Planck units may be There is a close sqrt natural number solution for Ω; constructed. r πe L30 2180π210Ω225P135 218π18P36 2154π154Ω165P99 Ω = = 2.0071 349 5432... (85) = / . e(e−1) M18T 22 V150 V18 V132 (70) 4.6. We can numerically solve the physical constants by re- 30 L 2 placing the mathematical (c∗, µ∗, R∗) with the CODATA mean 4π19 15 0 18 22 = (2 Ω ) (71) M T values for (c, µ0, R) as in section 3.8. We then find there is a combination of (c∗, µ∗, R∗) which reduces to h3 using the for- A6L6 218V18 236π42Ω45P27 214π14Ω15P9 0 = . / (72) mulas in 4.2. T 2 α6P18 V30 V12 r13 h∗ = 23π4Ω4 , u19 (86) A6L6 220π14Ω15 v5 = ( )2 (73) T 2 α3 2π10(µ∗)3 (h∗)3 = 0 , unit = u57 (87) 4.3. The electron function fe is both unit-less and non scalable 36(c∗)5α13(R∗)2 v0r0u0 = 1. It is therefore a natural (mathematical) constant. Likewise with the other dimensionful constants, we note that 3α2AL r3 these equations are equivalent to section 3.8; σ 7 π3α 5 , u−10 e = 2 = 2 3 Ω 2 (74) π v 4π5 e∗ 3 , unit u−81 ( ) = 4 = (88) σ3 (u−10)3 33(c∗) α8(R∗) f e π2 6 π2α 5 3, units e = = 4 (2 3 Ω ) = −30 = 1 (75) T u 5 ∗ 3 ∗ 3 π (µ0) 87 2 4 (k ) = , unit = u (89) 3α T v B ∗ 4 ∗ σ = P = 263π2αΩ5 , units = u20 (76) 332(c ) α5(R ) tp 6 2π r 3 ∗ ∗ 5 π (µ0) (u20)3 (G ) = , unit = u30 (90) f = t2σ3 = 4π2(263π2αΩ5)3, units = = 1 (77) 22036α11(R∗)2 e p tp (u30)2 16π10(R∗)(µ∗)3 4.4. The Sommerfeld fine structure constant alpha is a di- m∗ 3 0 , unit u45 ( e) = 8 = (91) mensionless mathematical constant. The following use a well 36(c∗) α7 known formula for alpha; 334π5(µ∗)3α19(R∗)2 r 3 0 , unit u24 ( d) = 3 ∗ 10 = (92) 5 2 6 5 (c ) 2h 32lpc α Q 1 α . πQ2 πl . . . α = 2 = 2 2 2 p 2 8 2 4 = (78) µ0e c π αQ 256lpc c 5 Unit u Here I assign a hypothetical β; unit = u and in this example α 128π4Ω3 α = 2(8π4Ω4)/( )( )2(2πΩ2) = α (79) use as scalars (r, v). 211π5Ω4 α √ 2πΩv 2πΩ 2πΩ v r13 1 v6 1 β = 2πΩa1/3 = = = √ ..., unit = u scalars = . . . = 1 (80) 2 2/15 1/5 5 7 6 r t k k v r r v (93) 19 1 u i = , unit = 1 units = = 1 (81) 15 u56(u−27)2u17 2π(2πΩ) r17 u8∗17 4.5. I have also premised a 2nd mathematical constant which j = , unit = = 1 I have denoted Omega. We can find a numerical solution us- v8 u17∗8 ∗ ∗ ∗ 3 6 3 3 3 ing the precise c , µ0, R ; 2 2 π Ω v A = β3( ) = , u3 (94) Ω = 2.0071349496...; α α r6 (c∗)35 (u17)35 β6 r5 , units = = 1 (82) G = ( j) = 23π4Ω6 , u6 (95) (µ∗)9(R∗)7 (u56)9(u13)7 23π2 v2 0 √ R = β8( i j) = Ω8r, u8 (96) ∗ 35 (c ) 2 35 α 9 1 7 = (2πΩ ) /( ) .( ) (83) 1 v5 (µ∗)9(R∗)7 211π5Ω4 22333π11α5Ω17 L−1 = 4πβ13(i j) = , u13 (97) 0 2π2Ω2 r9 ∗ 35 225 (c ) 4 Ω = , units = 1 (84) 15 2 r 15 295 21 157 ∗ 9 ∗ 7 26 M = 2πβ (i j ) = , u (98) 2 3 π (µ0) (R ) α v

5 5 Unit u virtual black-hole electron and the simulation hypothesis

P = β16(i j2) = Ωr2, u16 (99) 6 Virtual universe 17 2 2 17 V = β (i j ) = 2πΩ v, u (100) The electron formula fe can be constructed from ampere- 3 20 7 3 5 4 meters AL and time T and yet it is a dimensionless (math- ∗ 2 β 2 2 π Ω v 20 T = (i j ) = , u (101) ematical) constant; P πα α r6 f = (AL)3/T = 0.239x1023, units = 1 1 v6 e T −1 πβ30 i2 j3 , u30 This formula has 3 space dimensions L3 and 1 time dimen- = 2 ( ) = 9 (102) 2π r sion T. We could then speculate that if the vacuum of our 3-D π3αβ56 α µ∗ = (i4 j7) = r7, u56 (103) space is electro-magnetic in nature such that it is also a con- 0 23 211π5Ω4 struct of an ampere-meters (AL)3 ‘ether’ instead of an empty π3αβ90 α void measured in L3 meters, then the sum universe may also ∗−1 = (i6 j11) = v2r7, u90 (104) 0 23 29π3 be a dimensionless (mathematical) constant (aka a virtual uni- In SI units; verse) [27]; 3 √ funiverse = X(AL) /T, units = 1 v 1 v a1/3 = = = √ ... = 23326079.1...; unit = u In the “Trialogue on the number of fundamental physical r2 t2/15k1/5 k constants” was debated the number, from 0 to 3, of dimen- (105) sionful units required [5]. Here the answer is both 0 and 1; 0 r17 k17/4 u17∗8 = k2t = ... = 0.813x10−59, units = ... = 1 in that the electron, being a virtual particle, has no units, yet v8 v15/4 u8∗17 it can unfold into the Planck units and these can be decon- (106) structed in terms of the unit u = sqrt(velocity/mass), and so in Combining these unit=1 x, y ratios with unit u, we can derive terms of the physical universe, being a dimensioned universe the SI unit equivalents; (a universe of measurable units) the answer is 1. r r M9T 11 kg9 s11 x = , units = = 1 (107) L15 m15 7 Notes y = M2T, units = kg2 s = 1 (108) In 1963, Dirac noted regarding the fundamental constants; r V r m ”The physics of the future, of course, cannot have the three u, units = = (109) M kg.s quantities ~, e, c all as fundamental quantities. Only two of them can be fundamental, and the third must be derived from m3 u3 = = A (eq.6) those two.” [25] q3 s3 Q u6 ∗ y = m3/s2kg (G) In the article ”Surprises in numerical expressions of physical constants”, Amir et al write ... In science, as in life, ‘sur- (u8 ∗ y)2 ∗ x = q prises’ can be adequately appreciated only in the presence 13 −1 u ∗ x ∗ y = 1/m (lp ) of a null model, what we expect a priori. In physics, theo- 15 2 ries sometimes express the values of dimensionless physical u ∗ x ∗ y = kg (mP) constants as combinations of mathematical constants like pi 16 2 u ∗ x ∗ y = q (Q) or e. The inverse problem also arises, whereby the measured u17 ∗ x ∗ y2 = m/s (c) value of a physical constant admits a ‘surprisingly’ simple ap- proximation in terms of well-known mathematical constants. u20 ∗ x ∗ y2 = q5/kg4 (T , eq.8) P Can we estimate the probability for this to be a mere coinci- 30 2 3 −1 u ∗ x ∗ y = 1/s (tp ) dence? [24] 56 4 7 6 4 u ∗ x ∗ y = kg /q s (µ0, eq.11) J. Barrow and J. Webb on the fundamental constants; ’Some 90 6 11 56 17 17 −1 u ∗ x ∗ y = u ∗ u ∗ u (0 ) things never change. Physicists call them the constants of (u13 ∗ x ∗ y)3 nature. Such quantities as the velocity of light, c, Newton’s T/(AL)3; units = = x (110) u9 ∗ (u30 ∗ x2 ∗ y3) constant of gravitation, G, and the mass of the electron, me, are assumed to be the same at all places and times in the uni- (u20 ∗ x ∗ y2)3 T 2T 3 units /x verse. They form the scaﬀolding around which theories of P; = 30 2 3 2 = 1 (111) (u ∗ x ∗ y ) physics are erected, and they deﬁne the fabric of our uni- (u15 ∗ x ∗ y2)9 ∗ (u13 ∗ x ∗ y)15 verse. Physics has progressed by making ever more accu- M9T 11/L15; units = = x2 (u30 ∗ x2 ∗ y3)11 rate measurements of their values. And yet, remarkably, no (112) one has ever successfully predicted or explained any of the constants. Physicists have no idea why they take the special

6 6 Virtual universe virtual black-hole electron and the simulation hypothesis

numerical values that they do. In SI units, c is 299,792,458; wave function plays the role of an order parameter that sig- G is 6.673e-11; and me is 9.10938188e-31 -numbers that fol- nals a broken symmetry and the electron acquires an extended low no discernible pattern. The only thread running through space-time structure. Although speculative, this idea was cor- the values is that if many of them were even slightly different, roborated by a detailed analysis and calculation [8]. complex atomic structures such as living beings would not be possible. The desire to explain the constants has been one of Max Tegmark’s Mathematical Universe Hypothesis: Our the driving forces behind efforts to develop a complete uni- external physical reality is a mathematical structure. That is, fied description of nature, or ”theory of everything”. Physi- the physical universe is mathematics in a well-defined sense, cists have hoped that such a theory would show that each of and ”in those [worlds] complex enough to contain self-aware the constants of nature could have only one logically possi- substructures [they] will subjectively perceive themselves as ble value. It would reveal an underlying order to the seeming existing in a physically ’real’ world” [10]. arbitrariness of nature.’ [6]. Note: This article is an extension of an earlier article [26] At present, there is no candidate theory of everything that which has been incorporated as section 3. is able to calculate the mass of the electron [23]. References ”There are two kinds of fundamental constants of Nature: 1. Linnebo, Øystein, ”Platonism in the Philosophy of dimensionless (alpha) and dimensionful (c, h, G). To clarify Mathematics”, The Stanford Encyclopedia of Philos- the discussion I suggest to refer to the former as fundamen- ophy (Summer 2017 Edition), Edward N. Zalta (ed.), tal parameters and the latter as fundamental (or basic) units. plato.stanford.edu/archives/sum2017/entries/platonism- It is necessary and sufficient to have three basic units in or- mathematics der to reproduce in an experimentally meaningful way the di- 2. Nick Bostrom, Philosophical Quarterly 53 (211):243–255 mensions of all physical quantities. Theoretical equations de- (2003) scribing the physical world deal with dimensionless quantities and their solutions depend on dimensionless fundamental 3. https://en.wikipedia.org/wiki/Philosophy-of-mathematics parameters. But experiments, from which these theories are (22, Oct 2017) extracted and by which they could be tested, involve measure- 4. Leonardo Hsu, Jong-Ping Hsu; The physical basis of nat- ments, i.e. comparisons with standard dimensionful scales. ural units; Eur. Phys. J. Plus (2012) 127:11 Without standard dimensionful units and hence without cer- 5. Michael J. Duff et al JHEP03(2002)023 Trialogue on the tain conventions physics is unthinkable” -Trialogue [5]. number of fundamental constants 6. J. Barrow, J. Webb, Inconstant Constants, Scientific Amer- ”The fundamental constants divide into two categories, ican 292, 56 - 63 (2005) units independent and units dependent, because only the con- 7. M. Planck, Uber irreversible Strahlungsforgange. Ann. d. stants in the former category have values that are not deter- Phys. (4), (1900) 1, S. 69-122. mined by the human convention of units and so are true fundamental constants in the sense that they are inherent prop- 8. A. Burinskii, Gravitation and Cosmology, 2008, Vol. 14, erties of our universe. In comparison, constants in the latter No. 2, pp. 109–122; Pleiades Publishing, 2008. category are not fundamental constants in the sense that their DOI: 10.1134/S0202289308020011 particular values are determined by the human convention of 9. Feynman, Richard P. (1977). The Feynman Lectures on units” -L. and J. Hsu [4]. Physics, vol. I. Addison-Wesley. p. 22-10. ISBN 0-201- 02010-6. A charged rotating black hole is a black hole that pos- 10. Tegmark, Max (February 2008). ”The Mathematical Uni- sesses angular momentum and charge. In particular, it ro- verse”. Foundations of Physics. 38 (2): 101–150. tates about one of its axes of symmetry. In physics, there is a speculative notion that if there were a black hole with the 11. planckmomentum.com/calc/ (online calculator) same mass and charge as an electron, it would share many of 12. Magnetic monopole the properties of the electron including the magnetic moment en.wikipedia.org/wiki/Magnetic-monopole (10/2015) and Compton wavelength. This idea is substantiated within a 13. Fine structure constant series of papers published by Albert Einstein between 1927 en.wikipedia.org/wiki/Fine-structure-constant (06/2015) and 1949. In them, he showed that if elementary particles were treated as singularities in spacetime, it was unnecessary 14. Burinskii, A. (2005). ”The Dirac–Kerr electron”. to postulate geodesic motion as part of general relativity [26]. arXiv:hep-th/0507109 The Dirac Kerr–Newman black-hole electron was intro- 15. Rydberg constant duced by Burinskii using geometrical arguments. The Dirac http://physics.nist.gov/cgi-bin/cuu/Value?ryd

7 7 Notes virtual black-hole electron and the simulation hypothesis

16. Planck constant http://physics.nist.gov/cgi-bin/cuu/Value?ha 17. Electron mass http://physics.nist.gov/cgi-bin/cuu/Value?me 18. Fine structure constant http://physics.nist.gov/cgi-bin/cuu/Value?alphinv 19. Elementary charge http://physics.nist.gov/cgi-bin/cuu/Value?e 20. Vacuum of permeability http://physics.nist.gov/cgi-bin/cuu/Value?mu0 21. Gravitation constant http://physics.nist.gov/cgi-bin/cuu/Value?bg 22. Boltzmann constant http://physics.nist.gov/cgi-bin/cuu/Value?k 23. https://en.wikipedia.org/wiki/Theory-of-everything (02/2016) 24. Ariel Amir, Mikhail Lemeshko, Tadashi Tokieda; 26/02/2016 Surprises in numerical expressions of physical constants arXiv:1603.00299 [physics.pop-ph] 25. Dirac, Paul, The Evolution of the Physicist’s Picture of Nature, June 25, 2010 https://blogs.scientiﬁcamerican.com/guest-blog/the- evolution-of-the-physicists-picture-of-nature/ 26. Macleod, Malcolm, A virtual black-hole electron and the sqrt of Planck momentum, http://vixra.org/pdf/1102.0032v9.pdf 27. Macleod, Malcolm; 2017 edition (online), God the Pro- grammer, the philosophy behind a Virtual Universe, http://platoscode.com/

8 7 Notes