Physical constant (anomaly)

Anomalies within the dimensioned physical constants (G, h, c, e, me, kB) suggest a mathematical relationship linking the units (kg, m, s, A, K).


A dimensioned physical constant, sometimes denoted a fundamental physical constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. Common examples being the speed of light c, the gravitational constant G, the Planck constant h and the elementary charge e. These constants are usually measured in terms of SI units mass (kilogram), length (meter), time (second), charge (ampere), temperature (Kelvin) ... (kg, m, s, A, K ...).

These constants form the scaffolding around which the theories of physics are erected, and they define the fabric of our universe, but science has no idea why they take the special numerical values that they do, for these constants follow no discernible pattern. The desire to explain the constants has been one of the driving forces behind efforts to develop a complete unified description of nature, or "theory of everything". Physicists have hoped that such a theory would show that each of the constants of nature could have only one logically possible value. It would reveal an underlying order to the seeming arbitrariness of nature [1].

Notably a physical universe, as opposed to a mathematical universe (a computer simulation), has as a fundamental premise the concept that the universe scaffolding (of mass, space and time) exists, that somehow mass is, space is, time is ... these dimensions are real, and independent of each other ... we cannot measure distance in kilograms and amperes, or mass using length and temperature. The 2019 redefinition of SI base units resulted in 4 physical constants (h, c, e, kB) having independently assigned exact values (they cannot be derived in terms of each other), and this confirmed the independence of their associated SI units as shown in this table.

2019 redefinition of SI base units
constant SI units
Speed of light c
Planck constant h
Elementary charge e
Boltzmann constant kB


However there are anomalies which occur in certain combinations of the physical constants (G, h, c, e, me, kB) which suggest a mathematical relationship between the units (kg, m, s, A, K) [2]. In order for the dimensioned physical constants to be fundamental, the units must be independent of each other, there cannot be a unit number relationship ... however these anomalies question this fundamental assumption. Every combination predicted by the model returns an answer consistent with CODATA precision. Statistically therefore, can these anomalies be dismissed as coincidence?



Anomalies

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Unit number

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We can demonstrate a curious geometrical relationship between the units (kg, m, s, A) by selecting 2 dimensioned quantities, here are chosen r, v such that

 


The units (kg, m, s, A) remain independent of each (i.e.: the kg cannot be replaced by the m or the s ...), and so we still have 4 independent units, however if 3 (or more) units are combined together, in a specific ratio, they can cancel.

 

This f(X) embeds the units kg, m, s but itself is dimensionless, units = 1 (i.e.: it is a mathematical structure).


If we assign these SI units to the dimensioned quantities r, v;

units:  
units:  
units:   units = 1


Mass

 


Length (  = 1)

 
 
 
 


Time

 
 
 
 


We can also construct a unit-less structure using the ampere with length and time

 


If we assign a numerical value θ to r (θ = 8) and to v (θ = 17), then we can assign a unit number relationship to the SI units kg, m, s, A, K [3].

Table 1. unit relationship
attribute SI equivalent unit number θ scalars
(mass) kg 15  
(time) s -30  
(velocity) m/s 17  
(length) m -13  
(ampere) A 3  
(temperature) K 20  



Planck units

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The Planck units are direct measures of the SI units; Planck mass in kg, Planck length in m, Planck time in s ... and so they are analogues to the attributes in the above table. The SI Planck units have numerical values, however to derive a mathematical relation between these SI units we cannot use numerical values, this is because numerical values are simply dimensionless frequencies of the SI unit itself, 299792458 could refer to the speed of light 299792458m/s or equally the number of apples in a container, numbers such as 299792458 carry no unit-specific information, and so the units are treated as independent by default.

This can be resolved by assigning to each Planck unit a geometrical object (denoted MLTVA), and for which the geometry embeds the attribute (for example, the geometry of the time object T embeds the function time and so a descriptive s is not required). We may then combine these objects to form more complex objects; from electrons to galaxies, while still retaining the underlying attributes (of mass, wavelength, frequency …).

As this particular geometrical approach requires that the objects be interrelated (they are not independent of each other), this unit number relationship hypothesis can be easily tested. This is because, if these MLTVPA objects are natural Planck units, then they will be embedded within our dimensioned SI physical constants (G, h, c, e, me, KB...).


Table 2. is an example of object orientated units; it assigns MLTA objects as the geometry of 2 dimensionless physical constants; the Sommerfeld fine structure constant alpha and Omega. As alpha and Omega are dimensionless (alpha = 137.035999084, Omega = 2.0071349496), so too are these objects.


Table 2. MLTVA Geometrical objects
attribute geometrical object numerical
mass   1
time   3.14159265358...
velocity   25.3123819353...
length   79.5211931328...
ampere   234.18260736...


We can now solve those f(X) structures as dimensionless geometrical forms.

 

The electron fe (below) is one example of an f(X) structure


The dimensioned constants in terms of MLTA as the Planck units

Table 3. Physical constant unit numbers
SI constant geometrical analogue unit number θ
Speed of light c* = V 17
Planck constant   15+17-13=19
Gravitational constant   34-13-15=6
Elementary charge   3-30=-27
Boltzmann constant   17+15-3=29
Vacuum permeability   34+15+13-6=56


As Alpha and Omega can have (dimensionless) numerical values, we can use dimensioned numerical scalars to convert from the MLTVA objects to their SI equivalents.

For example, we can use scalar v to convert from dimensionless geometrical object V to dimensioned c.

scalar vSI = 11843707.905 m/s gives c = V*vSI = 25.3123819 * 11843707.905 m/s = 299792458 m/s (SI units)
scalar vimp = 7359.3232155 miles/s gives c = V*vimp = 186282 miles/s (imperial units)


As scalar v also carries the unit designation m/s or miles/s, then it is dimensioned, and so the unit number relationship θ applies to the scalar itself, for scalar v the unit number (θ = 17). Here each attribute is assigned a scalar.

Table 4. Scalars
attribute geometrical object scalar
mass   k (θ = 15)
time   t (θ = -30)
velocity   v (θ = 17)
length   l (θ = -13)
ampere   a (θ = 3)


Because the scalars follow the unit number relationship (units as uθ), we can find ratios where the scalars cancel. Here are examples (units = 1), as such only 2 scalars are required, for example, if we know the numerical value for a and for l then we know the numerical value for t (t = a3l3), and from l and t we know the value for k.

 


This means that once any 2 scalars have been assigned values, the other scalars are then defined by default, consequently the CODATA 2014 values are used here as only 2 constants (c, μ0) are assigned exact values, following the 2019 redefinition of SI base units 4 constants have been independently assigned exact values which is problematic in terms of this model. Here the scalars are each defined in terms of the 2 scalars r, v because these can be derived from the 2 exact constants (c, μ0).

 
 


Table 5. Geometrical objects
attribute geometrical object unit number θ scalar r(8), v(17)
mass   15 = 8*4-17  
time   -30 = 8*9-17*6  
velocity   17 v
length   -13 = 8*9-17*5  
ampere   3 = 17*3-8*6  


Table 6. Comparison θ; SI units and scalars
constant θ from SI units MLTVA θ from r(8), v(17)
c   (-13+30 = 17) c* =   17
h   (15-26+30=19) h* =   8*13-17*5=19
G   (-39-15+60=6) G* =   8*5-17*2=6
e   (3-30=-27) e* =   8*3-17*3=-27
kB   (15-26+60-20=29) kB* =   8*10-17*3=29
μ0   (15-13+60-6=56) μ0* =   8*7=56




Calculating the electron

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The electron object (formula fe) is a mathematical particle (units and scalars cancel).

  units = 1


In this example, embedded within the electron are the objects for charge, length and time ALT. AL as an ampere-meter (ampere-length) are the units for a magnetic monopole.

 
 
 


Electron parameters

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Associated with the electron are dimensioned parameters, these parameters however are a function of the MLTA units, the formula fe dictating the frequency of these units. By setting MLTA to their SI Planck unit equivalents;

electron mass   (M = Planck mass =  

electron wavelength   (L = Planck length =  

elementary charge   (T = Planck time) =  

Rydberg constant  





Calculating from (α, Ω)

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If we can reduce the 5 SI units to 2 scalars (example; r, v in tables 5, 6), then we can find combinations of the physical constants (G, h, c, e, me, kB) where the unit numbers θ and the scalars will cancel, these combinations, which are unit-less (units = 1), will then return the same numerical value as the MLTVA object equivalents. This is because if the scalars have cancelled, and as the scalars embed the SI conversion values as well as the SI units, then these combinations are defaulting to the underlying MLTVA objects (the SI component has cancelled).

This should therefore apply to any set of units, even extraterrestrial and non-human ones, suggesting that these MLTVA objects could be 'natural' units, the precision of the results in following table can be used to verify this conjecture.

...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck [4][5]


For example;

  0.228 473 759... 10-58
  0.228 473 639... 10-58


Here we solve physical constant combinations using only α, Ω (and the mathematical constants 2, 3, π). As the scalars (v, r) have cancelled, we do not need to know their values or the units. This means that column 1 does not equal column 2, rather column 1 (sans scalars) is column 2. The precision of the results depends on the precision of the SI constants; combinations with G and kB return the least precise values.

Note: the geometry   (integer n ≥ 0) is common to all ratios where units and scalars cancel


Table 7. Dimensionless combinations (α, Ω)
CODATA 2014 (mean) (α, Ω) units uΘ = 1 scalars = 1
  1.000 8254   = 1.0    
  0.228 473 639... 10-58   0.228 473 759... 10-58    
  0.326 103 528 6170... 10301   0.326 103 528 6170... 10301    
  0.170 514 342... 1092   0.170 514 368... 1092    
  73 095 507 858.   73 035 235 897.    
  3.376 716   3.381 506    





Calculating from (α, Ω, v, r)

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Here the attributes are defined in terms of 2 scalars; from c (exact value) is v (θ = 17), and from μ0 (exact value) we can derive r (θ = 8), hence the rationale for choosing scalars r and v in table 1.


Table 8. Dimensioned constants (α, Ω, v, r)
constant geometrical object calculated (α, Ω, r, v) CODATA 2014 [6]
Planck constant   6.626 069 134 e-34, u19 6.626 070 040(81) e-34
Gravitational constant   6.672 497 192 29 e11, u6 6.674 08(31) e-11
Elementary charge   1.602 176 511 30 e-19, u-27 1.602 176 620 8(98) e-19
Boltzmann constant   1.379 510 147 52 e-23, u29 1.380 648 52(79) e-23
Vacuum permeability   4π/10^7, u56 4π/10^7 (exact)






Calculating from (c, R, μ0, α)

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By matching the unit numbers we can numerically solve the least precise dimensioned physical constants (G, h, e, me, kB ...) using the 3 most precise (CODATA 2014); speed of light c (exact value), vacuum permeability μ0 (exact value), Rydberg constant R (12-13 digits) and the dimensionless fine structure constant alpha.

R = 10973731.568508 (θ=13)

c = 299792458 (θ=17)

μ0 = 4π/107 (θ=56)

α = 137.035999139 (θ=0)


For example

 
 



Table 10. R, c, μ0, α ... (CODATA 2014 mean)
constant formula* calculated θ CODATA 2014 [7] Units
Planck constant   h* = 6.626 069 134 e-34  , 15*3-3*6+30 = 57 h = 6.626 070 040(81) e-34  , θ = 15-13*2+30 = 19
Gravitational constant   G* = 6.672 497 192 29 e11  , 15-13*3-3*2+30*2 = 30 G = 6.674 08(31) e-11  , θ = -13*3-15+30*2 = 6
Elementary charge   e* = 1.602 176 511 30 e-19  , -30*4+13*3 = -81 e = 1.602 176 620 8(98) e-19  , θ = 3-30 = -27
Boltzmann constant   kB* = 1.379 510 147 52 e-23  , 15*3+30*2-3*6 = 87 kB = 1.380 648 52(79) e-23  , θ = 15-26+60-20 = 29
Electron mass   me* = 9.109 382 312 56 e-31, u = 15  , 15*3-30*2+13*6-3*6 = 45 me = 9.109 383 56(11) e-31  , θ = 15
  1.0  , 13*2-3*2-20 = 0 5kB/(27α6e2mec4) = 1.000 825  , θ = 0
Gyromagnetic ratio   e*/2π) = 28024.953 55  , -13*3-30*2+3*6-15*3 = -126 γe/2π = 28024.951 64(17)  , θ = -42
Planck length   lp* = 0.161 603 660 096 e-34  , 15*9-30*17+13*18-3*18 = -195 lp = 0.161 622 9(38) e-34  , θ = -13
Planck mass   mP* = 0.217 672 817 580 e-7 u =  , 15*6-13*3+30*7-3*12 = 225 mP = 0.217 647 0(51) e-7  , θ = 15
  1.0  , -13*4-15*3+30*4-3-20 = 0 211π3G2kB/(α2hc2emP) = 1.001 418  , θ = 0





Calculating from (α)

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Combinations which reduce to the dimensionless (no scalars or units) fine structure constant. For example;

 
 
 


Table 9. fine structure constant
CODATA 2014 geometrical (α)
  137.035 999 139(31)  
  137.133 167 47  
  137.054 833 44  
  137.119 576 89  





Table of constants

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We can construct a table of constants using these 3 geometries. Setting

 , units =   = 1 (unit number θ = (-13*15) - (15*9) - (-30*11) = 0, no scalars)
  , units =   = u1 = u (unit number = -13 -15 +30 = 2/2 = 1, with scalars v, r)
  , units =   = 1, (unit number = 15*2 -30 = 0, with scalars v, r)


Table 9. Table of Constants
Constant θ Geometrical object (α, Ω, v, r) Unit Calculated CODATA 2014
Time (Planck)       T = 5.390 517 866 e-44 tp = 5.391 247(60) e-44
Elementary charge       e* = 1.602 176 511 30 e-19 e = 1.602 176 620 8(98) e-19
Length (Planck)       L = 0.161 603 660 096 e-34 lp = 0.161 622 9(38) e-34
Ampere       A = 0.297 221 e25 e/tp = 0.297 181 e25
Gravitational constant       G* = 6.672 497 192 29 e11 G = 6.674 08(31) e-11
Mass (Planck)       M = .217 672 817 580 e-7 mP = .217 647 0(51) e-7
Velocity       V = 299 792 458 c = 299 792 458
Planck constant       h* = 6.626 069 134 e-34 h = 6.626 070 040(81) e-34
Planck temperature       Tp* = 1.418 145 219 e32 Tp = 1.416 784(16) e32
Boltzmann constant       kB* = 1.379 510 147 52 e-23 kB = 1.380 648 52(79) e-23
Vacuum permeability       μ0* = 4π/10^7 μ0 = 4π/10^7




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References

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  1. J. Barrow, J. Webb "Inconsistent constants". Scientific American 292: 56. 2005. 
  2. Macleod, M.J. "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus 113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x. 
  3. Are these physical constant anomalies evidence of a mathematical relation between the SI units?. doi:10.13140/RG.2.2.15874.15041/6. "
  4. Planck (1899), p. 479.
  5. *Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System", 287–296.
  6. [1] | CODATA, The Committee on Data for Science and Technology | (2014)
  7. [2] | CODATA, The Committee on Data for Science and Technology | (2014)